0213_0321_1230_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 119, 257, 524, 1023, 1936, 3582, 6522, 11746, 21005, 37406, 66479, 118095, 209923, 373673, 666392, 1190939, 2133156, 3829446, 6889790, 12421942, 22440441, 40613486, 73627891, 133685451, 243071535, 442520921, 806544908, 1471525095, 2687228664, 4911305350, 8982682826, 16439824490, 30105019109, 55157313286, 101103120511, 185396171127, 340088251771, 624049870593, 1145428226600, 2102922565139, 3861650764140, 7092607822862, 13029037466238, 23937758575758, 43985722638865, 80833299910550, 148563881057619, 273070784693315, 501962946599559, 922780474289201, 1696492048368012, 3119096173885167, 5734907243854848, 10544894717941198, 19389835934713114, 35654974947254226, 65565980449559005, 120572403231788654, 221731245378374479, 407769127709609663, 749910161719278099, 1379147418856492569, 2536400977733945528, 4664769711807333099, 8579203531343766228, 15778570797328461014, 29019626039868766782, 53372678944373402630, 98163236356792021961, 180543180492087758206, 332059095519527897779, 610733151245735706011, 1123283065860953840639, 2065990590107147675785, 3799869723298369645836, 6988921572831933518871, 12854422995887445310504, 23642633595233204979670, 43485038576818034448874, 79980574884019591535898, 147105787185017188040565, 270567420490882077524310, 497647342533892477288095, 915310130028792626137383, 1683508032846541684005195, 3096438225021202173339377, 5695212247301486371999816, 10475087084186155503309219, 19266621994930719210653532, 35436734343857161521438750, 65178140878834711832372478, 119881007690922068596384670, 220495090842774093578542241, 405552957814990501667003430, 745926982680306443129400467
Generating function in Maple syntax:
(x^11+x^10-4*x^9-4*x^8+7*x^7-5*x^5-x^4-4*x^3+9*x^2-5*x+1)/(x^2+x-1)/(x^3+x^2+x-1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{11}+x^{10}-4 x^{9}-4 x^{8}+7 x^{7}-5 x^{5}-x^{4}-4 x^{3}+9 x^{2}-5 x +1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**11 + x**10 - 4*x**9 - 4*x**8 + 7*x**7 - 5*x**5 - x**4 - 4*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**4*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^3+x^2+x-1)*(x-1)^4*F(x)+x^11+x^10-4*x^9-4*x^8+7*x^7-5*x^5-x^4-4*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{11}+x^{10}-4 x^{9}-4 x^{8}+7 x^{7}-5 x^{5}-x^{4}-4 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/264*(((-21*11^(1/2)+99*I)*3^(1/2)-63*I*11^(1/2)+99)*(17+3*11^(1/2)*3^(1/2))^(1/3)+264+((15*11^(1/2)+99*I)*3^(1/2)-45*I*11^(1/2)-99)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/264*(((15*11^(1/2)-99*I)*3^(1/2)+45*I*11^(1/2)-99)*(17+3*11^(1/2)*3^(1/2))^(2/3)+264+((-21*11^(1/2)-99*I)*3^(1/2)+63*I*11^(1/2)+99)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/264*((-30*11^(1/2)*3^(1/2)+198)*(17+3*11^(1/2)*3^(1/2))^(2/3)+42*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-198*(17+3*11^(1/2)*3^(1/2))^(1/3)+264)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/264*(-660*5^(1/2)+1452)*(-1/2*5^(1/2)-1/2)^(-n)+1/264*(660*5^(1/2)+1452)*(1/2*5^(1/2)-1/2)^(-n)-1/3*n^3-3/2*n^2-37/6*n-16)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(-21 \sqrt{11}+99 \,\mathrm{I}\right) \sqrt{3}-63 \,\mathrm{I} \sqrt{11}+99\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+264+\left(\left(15 \sqrt{11}+99 \,\mathrm{I}\right) \sqrt{3}-45 \,\mathrm{I} \sqrt{11}-99\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{264}\\+\\\frac{\left(\left(\left(15 \sqrt{11}-99 \,\mathrm{I}\right) \sqrt{3}+45 \,\mathrm{I} \sqrt{11}-99\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+264+\left(\left(-21 \sqrt{11}-99 \,\mathrm{I}\right) \sqrt{3}+63 \,\mathrm{I} \sqrt{11}+99\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{264}\\+\\\frac{\left(\left(-30 \sqrt{11}\, \sqrt{3}+198\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+42 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-198 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+264\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{264}\\+\frac{\left(-660 \sqrt{5}+1452\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{264}+\frac{\left(660 \sqrt{5}+1452\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{264}-\\\frac{n^{3}}{3}-\frac{3 n^{2}}{2}-\frac{37 n}{6}-16 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 119
a(7) = 257
a(8) = 524
a(9) = 1023
a(10) = 1936
a(11) = 3582
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)-1/3*(-2+n)*(2*n+1)*(n+3), n >= 12
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) = 51
a \! \left(6\right) = 119
a \! \left(7\right) = 257
a \! \left(8\right) = 524
a \! \left(9\right) = 1023
a \! \left(10\right) = 1936
a \! \left(11\right) = 3582
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)-\frac{\left(-2+n \right) \left(2 n +1\right) \left(n +3\right)}{3}, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20510/
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[17,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[18,x]+F[29,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[18,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[18,x]
F[29,x] = F[30,x]+F[31,x]+F[55,x]
F[30,x] = 0
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[38,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[15,x]+F[30,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[14,x]
F[38,x] = F[29,x]+F[39,x]
F[39,x] = 2*F[30,x]+F[40,x]+F[53,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[46,x]+F[7,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[45,x]
F[49,x] = F[42,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[42,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[27,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[18,x]+F[42,x]
F[58,x] = F[27,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_{17}\! \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_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right)
F_{38}\! \left(x \right) = F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{53}\! \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_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{7}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{27}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{58}\! \left(x \right) = F_{27}\! \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_17(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))
Eq(F_17(x), F_18(x) + F_29(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_18(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_18(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_55(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_38(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_15(x) + F_30(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_14(x))
Eq(F_38(x), F_29(x) + F_39(x))
Eq(F_39(x), 2*F_30(x) + F_40(x) + F_53(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_46(x) + F_7(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_42(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_42(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_27(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_18(x) + F_42(x))
Eq(F_58(x), F_27(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:
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