0231_0321_1032_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 52, 129, 304, 696, 1567, 3495, 7755, 17159, 37909, 83684, 184654, 407361, 898568, 1981971, 4371504, 9641798, 21265811, 46903393, 103448875, 228163877, 503231489, 1109912222, 2447988718, 5399209351, 11908331380, 26264651965, 57928513800, 127765359532, 281795371615, 621519257651, 1370803875491, 3023403123291, 6668325504965, 14707454886192, 32438312896486, 71544951298789, 157797357484664, 348033027866751, 767611007033272, 1693019371552234, 3734071770972291, 8235754548978973, 18164528469511347, 40063128709996201, 88362011968972641, 194888552407457946, 429840233524913462, 948042479018800987, 2090973510445061396, 4611787254415037785, 10171616987848878144, 22434207486142819328, 49480202226700678143, 109132021441250236191, 240698250368643293531, 530876702963987267087, 1170885427369224772309, 2582469105107092840156, 5695814913178172949470, 12562515253725570673385, 27707499612558234189128, 61110814138294641329995, 134784143530314853335712, 297275786673187940862958, 655662387484670523058387, 1446108918499655899455033, 3189493623672499739775643, 7034649634829670002612365, 15515408188158995904682529, 34220309999990491549143542, 75475269634810653100902366, 166465947457780302106490255, 367152204915551095762127124, 809779679465912844625159765, 1786025306389605991356813016, 3939202817694763078475756468, 8688185314855439001576676095, 19162395936100483994510168683, 42263994689895731067496097395, 93216174694646901136568874531, 205594745325394286267647921477, 453453485340684303602791944168, 1000123145376015508342152766774, 2205841036077425302951953459021, 4865135557495534909506698866296, 10730394260367085327355550503543, 23666629556811595957663054470376, 52198394671118726824832807811410, 115127183602604538977021166130819, 253920996762020673911705386736565, 560040388195160074648243581289187, 1235207959992924688273508328713937, 2724336916747870050458722044169281, 6008714221690900175565687669632690, 13252636403374725039404883667984358, 29229609723497320129268489380143139
Generating function in Maple syntax:
-(3*x^5-4*x^4+3*x^3-6*x^2+4*x-1)/(x^3+2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{3 x^{5}-4 x^{4}+3 x^{3}-6 x^{2}+4 x -1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-3*x**5 + 4*x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+2*x-1)*(x-1)^3*F(x)+3*x^5-4*x^4+3*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{5}-4 x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
121/45312*((-11328/121*(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-n)*(n^2+8*n-3/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*576^n*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+1196/363*(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)+48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((6*I*59^(1/2)-18)*18^(1/3)-54*I*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(59^(1/2)*3^(2*n+1/2)*64^n*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)+3481/598*(3^(2*n+2/3)*2^(6*n+1/3)*(9+59^(1/2)*3^(1/2))^(2/3)-42/59*3^(2*n+1/3)*2^(6*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3))*(108+12*59^(1/2)*3^(1/2))^(-2/3*n))*(-8*2^(2/3)*(I*3^(5/6)+3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)+2*2^(1/3)*(3^(1/6)*59^(1/2)-3*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3)+(12*I-4*3^(1/2))*59^(1/2)-36*I*3^(1/2)+36)^(-n)+(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-n)*(-826/121*(I+299/1239*59^(1/2))*64^n*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)-3481/726*(I-121/3481*59^(1/2))*64^n*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)+((-3481/363+I*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3)*3^(2*n+2/3)*2^(6*n+1/3)+598/121*(I*59^(1/2)+413/299)*3^(2*n+1/3)*(9+59^(1/2)*3^(1/2))^(1/3)*2^(6*n+2/3)+9440/121*(1+((8*(108+12*59^(1/2)*3^(1/2))^(1/3)-8*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-I*59^(1/2)+3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)-2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))/(-16*(108+12*59^(1/2)*3^(1/2))^(1/3)+(2*2^(1/3)*3^(1/6)*59^(1/2)-6*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3)))^n)*576^n)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)))*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(1/3*64^n*59^(1/2)*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)*(-8*2^(2/3)*(I*3^(5/6)+3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)+2*2^(1/3)*(3^(1/6)*59^(1/2)-3*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3)+(12*I-4*3^(1/2))*59^(1/2)-36*I*3^(1/2)+36)^(-n)*(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)-48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-6*I*59^(1/2)-18)*18^(1/3)+54*I*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-n)*(-826/121*64^n*(I-299/1239*59^(1/2))*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)-3481/726*64^n*(I+121/3481*59^(1/2))*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)+(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*((I*59^(1/2)+3481/363)*(9+59^(1/2)*3^(1/2))^(2/3)*3^(2*n+2/3)*2^(6*n+1/3)+598/121*3^(2*n+1/3)*(9+59^(1/2)*3^(1/2))^(1/3)*(I*59^(1/2)-413/299)*2^(6*n+2/3)-9440/121*576^n))))*((3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-9*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+48*I*3^(1/2)-48)^(-n)
Explicit closed form in latex syntax:
\frac{121 \left(\left(-\frac{11328 \left(\frac{\left(-3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{3^{\frac{5}{6}} \left(i+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{i \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(n^{2}+8 n -\frac{3}{2}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} 576^{n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(i \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}}{121}+\frac{1196 \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(6 i \sqrt{59}-18\right) 18^{\frac{1}{3}}-54 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\sqrt{59}\, 3^{2 n +\frac{1}{2}} 64^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3481 \left(3^{2 n +\frac{2}{3}} 2^{6 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{42 \,3^{2 n +\frac{1}{3}} 2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{59}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{598}\right) \left(-8 \,2^{\frac{2}{3}} \left(i 3^{\frac{5}{6}}+3^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+2 \,2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \sqrt{59}-3 \,3^{\frac{2}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(12 i-4 \sqrt{3}\right) \sqrt{59}-36 i \sqrt{3}+36\right)^{-n}}{363}+\left(\frac{\left(-3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{3^{\frac{5}{6}} \left(i+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{i \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(-\frac{826 \left(i+\frac{299 \sqrt{59}}{1239}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{121}-\frac{3481 \left(i-\frac{121 \sqrt{59}}{3481}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{726}+\left(\left(-\frac{3481}{363}+i \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 3^{2 n +\frac{2}{3}} 2^{6 n +\frac{1}{3}}+\frac{598 \left(i \sqrt{59}+\frac{413}{299}\right) 3^{2 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 2^{6 n +\frac{2}{3}}}{121}+\frac{9440 \left(1+\left(\frac{8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-i \sqrt{59}+3\right) 18^{\frac{1}{3}}+9 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{-16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}-6 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}\right)^{n}\right) 576^{n}}{121}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}\right)\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-i \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}-\left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(i \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(\frac{64^{n} \sqrt{59}\, 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(-8 \,2^{\frac{2}{3}} \left(i 3^{\frac{5}{6}}+3^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+2 \,2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \sqrt{59}-3 \,3^{\frac{2}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(12 i-4 \sqrt{3}\right) \sqrt{59}-36 i \sqrt{3}+36\right)^{-n} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 i \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{3}+\left(\frac{\left(-3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{3^{\frac{5}{6}} \left(i+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{i \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(-\frac{826 \,64^{n} \left(i-\frac{299 \sqrt{59}}{1239}\right) 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{121}-\frac{3481 \,64^{n} \left(i+\frac{121 \sqrt{59}}{3481}\right) 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{726}+\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(i \sqrt{59}+\frac{3481}{363}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 3^{2 n +\frac{2}{3}} 2^{6 n +\frac{1}{3}}+\frac{598 \,3^{2 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(i \sqrt{59}-\frac{413}{299}\right) 2^{6 n +\frac{2}{3}}}{121}-\frac{9440 \,576^{n}}{121}\right)\right)\right)\right) \left(\left(3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \left(i-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 i \sqrt{3}-48\right)^{-n}}{45312}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 52
a(n) = -1/2*n^2-2*a(n+2)+a(n+3)-9/2*n-1, n >= 6
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 52
a \! \left(n \right) = -\frac{n^{2}}{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-\frac{9 n}{2}-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20769/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[18,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[17,x]
F[17,x] = F[15,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[20,x]+F[21,x]+F[46,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[20,x]+F[25,x]+F[27,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[23,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]+F[17,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[20,x]+F[21,x]+F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[42,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[40,x]
F[50,x] = F[36,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[17,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[54,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{18}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{46}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{42}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)
F_{50}\! \left(x \right) = F_{36}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{17}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_18(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_17(x))
Eq(F_17(x), F_15(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_46(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_20(x) + F_25(x) + F_27(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_23(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x) + F_17(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_20(x) + F_21(x) + F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_42(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_40(x))
Eq(F_50(x), F_36(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_17(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_54(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 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