0231_1203_1230_2301_3012
Counting sequence:
1, 1, 2, 6, 19, 56, 151, 380, 914, 2142, 4959, 11446, 26487, 61637, 144421, 340785, 809497, 1934369, 4646445, 11210750, 27150820, 65963257, 160680778, 392259690, 959327138, 2349659725, 5761999889, 14144121479, 34748348882, 85424560478, 210120414945, 517067019369, 1272865926092, 3134342349605, 7719945836482, 19018079415739, 46858435630367, 115468823179647, 284568633298174, 701368285187681, 1728761565841568, 4261360981451786, 10504641440470832, 25895846528822160, 63839863810699030, 157385350826986890, 388012031654776004, 956605779544068825, 2358448643639057941, 5814661034311727774, 14335936782706971557, 35345224940232840169, 87144073050424668855, 214855726048841513198, 529733710164313650584, 1306079422365066075706, 3220198302386710861859, 7939561199417434790371, 19575419495237344863116, 48264322775834261918800, 118998588995026475682841, 293398424125894323941014, 723392566934443095869521, 1783571608250370976708750, 4397514259020932971473024, 10842367472263169434699268, 26732594971955052101490721, 65911048410079012727230720, 162508247410354003866771325, 400675384932933518458362131, 987893152762295913327787935, 2435719838319767713913289603, 6005438563674771126977242407, 14806832160203892243502471575, 36507290615034533611217472479, 90011307905410020868997922792, 221929256137613160671829925114, 547182319089574276218819773831, 1349116843828597638208798739044, 3326343397163631201927937882172, 8201336024548804541666922271904, 20220977039399945264339807266995, 49856257017420832340179255304931, 122924148528478377434760400871329, 303078235309305905372648048444328, 747260959975687674420294556114604, 1842425089598843189288152387898587, 4542630225759225873500947324018251, 11200178257598498115487643841901882, 27614837015651887751637761546903147, 68086347142407728228270600223363492, 167871737645196948858666387232969877, 413899724677764778262845557554433301, 1020499249436417474872101544922406157, 2516113582340676697608656454017441704, 6203657246310393436579612564925499699, 15295558802040402054325388225993361518, 37712289684017589870097290644623516672, 92982336385571367004236721549756901714, 229254573369862399800907922466799171754, 565243480286732883735682603678103757740
Generating function in Maple syntax:
-(4*x^9+4*x^8-48*x^7+126*x^6-186*x^5+177*x^4-111*x^3+44*x^2-10*x+1)/(2*x-1)/(3*x^3-5*x^2+4*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{4 x^{9}+4 x^{8}-48 x^{7}+126 x^{6}-186 x^{5}+177 x^{4}-111 x^{3}+44 x^{2}-10 x +1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-4*x**9 - 4*x**8 + 48*x**7 - 126*x**6 + 186*x**5 - 177*x**4 + 111*x**3 - 44*x**2 + 10*x - 1)/((x - 1)**5*(2*x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(3*x^3-5*x^2+4*x-1)*(x-1)^5*F(x)+4*x^9+4*x^8-48*x^7+126*x^6-186*x^5+177*x^4-111*x^3+44*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+4 x^{9}+4 x^{8}-48 x^{7}+126 x^{6}-186 x^{5}+177 x^{4}-111 x^{3}+44 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/270072*(-589*((I-51/589*31^(1/2))*3^(1/2)-153/589*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)+60016+682*((I-6/31*31^(1/2))*3^(1/2)+18/31*I*31^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3))*(47/8712*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/270072*(-682*2^(2/3)*((I+6/31*31^(1/2))*3^(1/2)+18/31*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(1/3)+60016+589*2^(1/3)*((I+51/589*31^(1/2))*3^(1/2)-153/589*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*2^(1/3)*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/270072*((264*3^(1/2)*2^(2/3)*31^(1/2)+1364*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+60016+(-102*31^(1/2)*3^(1/2)*2^(1/3)+1178*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/24*n^4-7/12*n^3+23/24*n^2-41/12*n+2^(n+1)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-589 \left(\left(\mathrm{I}-\frac{51 \sqrt{31}}{589}\right) \sqrt{3}-\frac{153 \,\mathrm{I} \sqrt{31}}{589}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+60016+682 \left(\left(\mathrm{I}-\frac{6 \sqrt{31}}{31}\right) \sqrt{3}+\frac{18 \,\mathrm{I} \sqrt{31}}{31}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{47 \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{270072}\\+\\\frac{\left(-682 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{6 \sqrt{31}}{31}\right) \sqrt{3}+\frac{18 \,\mathrm{I} \sqrt{31}}{31}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+60016+589 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{51 \sqrt{31}}{589}\right) \sqrt{3}-\frac{153 \,\mathrm{I} \sqrt{31}}{589}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{270072}\\+\\\frac{\left(\left(264 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{31}+1364 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+60016+\left(-102 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+1178 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{270072}\\+\frac{n^{4}}{24}-\frac{7 n^{3}}{12}+\frac{23 n^{2}}{24}-\frac{41 n}{12}+2^{n +1}-1 & \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) = 56
a(6) = 151
a(7) = 380
a(8) = 914
a(9) = 2142
a(n+4) = -6*a(n)+13*a(n+1)-13*a(n+2)+6*a(n+3)+1/24*(n-1)*(n^3-17*n^2+54*n-144), n >= 10
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) = 56
a \! \left(6\right) = 151
a \! \left(7\right) = 380
a \! \left(8\right) = 914
a \! \left(9\right) = 2142
a \! \left(n +4\right) = -6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)+\frac{\left(n -1\right) \left(n^{3}-17 n^{2}+54 n -144\right)}{24}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22034/
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[14,x]*F[4,x]
F[4,x] = F[30,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[5,x]+F[8,x]
F[8,x] = F[14,x]*F[9,x]
F[9,x] = F[10,x]+F[15,x]
F[10,x] = F[11,x]^2
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[16,x]
F[16,x] = F[11,x]^2*F[14,x]*F[17,x]
F[17,x] = F[18,x]+F[25,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[12,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[24,x]
F[21,x] = 0
F[22,x] = F[14,x]*F[23,x]
F[23,x] = F[12,x]+F[20,x]
F[24,x] = F[14,x]*F[19,x]
F[25,x] = F[12,x]+F[26,x]
F[26,x] = F[21,x]+F[27,x]+F[29,x]
F[27,x] = F[14,x]*F[28,x]
F[28,x] = F[12,x]+F[26,x]
F[29,x] = F[14,x]*F[25,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]*F[32,x]
F[32,x] = F[33,x]+F[9,x]
F[33,x] = F[34,x]
F[34,x] = F[11,x]^2*F[14,x]*F[35,x]
F[35,x] = F[36,x]+F[46,x]
F[36,x] = F[11,x]*F[37,x]*F[40,x]
F[37,x] = F[1,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[14,x]*F[37,x]*F[40,x]
F[40,x] = F[1,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[14,x]*F[43,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[11,x]^3*F[14,x]
F[46,x] = F[11,x]*F[12,x]*F[47,x]
F[47,x] = F[12,x]+F[37,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{30}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11} \left(x \right)^{2}
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{46}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right) F_{37}\! \left(x \right) F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11} \left(x \right)^{3} F_{14}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right)+F_{37}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_30(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_5(x) + F_8(x))
Eq(F_8(x), F_14(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_15(x))
Eq(F_10(x), F_11(x)**2)
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_11(x)**2*F_14(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_25(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_12(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_24(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_14(x)*F_23(x))
Eq(F_23(x), F_12(x) + F_20(x))
Eq(F_24(x), F_14(x)*F_19(x))
Eq(F_25(x), F_12(x) + F_26(x))
Eq(F_26(x), F_21(x) + F_27(x) + F_29(x))
Eq(F_27(x), F_14(x)*F_28(x))
Eq(F_28(x), F_12(x) + F_26(x))
Eq(F_29(x), F_14(x)*F_25(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_9(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_11(x)**2*F_14(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_46(x))
Eq(F_36(x), F_11(x)*F_37(x)*F_40(x))
Eq(F_37(x), F_1(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_14(x)*F_37(x)*F_40(x))
Eq(F_40(x), F_1(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_14(x)*F_43(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_11(x)**3*F_14(x))
Eq(F_46(x), F_11(x)*F_12(x)*F_47(x))
Eq(F_47(x), F_12(x) + F_37(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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