0123_0132_0231_1320_2013
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 368, 911, 2204, 5250, 12382, 29025, 67792, 158006, 367836, 855757, 1990180, 4627558, 10758904, 25012763, 58149198, 135182322, 314262894, 730575029, 1698383668, 3948266164, 9178608810, 21337680721, 49604094010, 115315533080, 268076096030, 623201320465, 1448767307406, 3367975382806, 7829592860088, 18201595128637, 42313575055716, 98367127777768, 228675828358982, 531606804586303, 1235835884828632, 2872969873682972, 6678844655989476, 15526430105443499, 36094570878047804, 83909697079260124, 195066379587143488, 453474315480974281, 1054199884347913334, 2450717401668953400, 5697226751792027462, 13244445336386109795, 30789599907243250012, 71577135800749581386, 166396652924148378786, 386825287078189249887, 899259691187020601250, 2090525152328831712078, 4859881361690643214534, 11297853471601286853263, 26264322843751405877078, 61057142949742287175292, 141940636633325336662864, 329971946844242841554859, 767091710215820138555868, 1783271873592300069264494, 4145604147189502772282430, 9637360404599728316922141, 22404144793012479475516240, 52083317717027484565039330, 121079023969657223061062460, 281474581267929179528682185, 654349013581500577029020676, 1521176902178300595090822386, 3536307260639829810743172888, 8220916991144388819076962167, 19111313354331807430835433486, 44428413341346474465095622238, 103283530306520197352693038062, 240105077591199450558723382353, 558176585501904431435879775412, 1297603131629834590542885684992, 3016561301476894359315620975146, 7012654226672918328296971422829, 16302443208694800856802558098554, 37898582474215460273129352524004, 88103515231929697434079912900158, 204815823956052972612783591857101, 476139023878514983243320302404814, 1106888938955368701938473636514050, 2573204593065129111941563896707192, 5981974925163164915191064719619865, 13906404528314605223628540002084932, 32328468327682614952445054463852116, 75154570851581799325269148107140486, 174713180427694773294545875395940667, 406158867907603336185544384437536120, 944204813719002261292810550628015048, 2195009885769494884801888758405065764, 5102778897778083468005589558396860263
Generating function in Maple syntax:
-(x^8-4*x^7+7*x^6-9*x^5+15*x^4-19*x^3+15*x^2-6*x+1)/(x^3-2*x^2+3*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{x^{8}-4 x^{7}+7 x^{6}-9 x^{5}+15 x^{4}-19 x^{3}+15 x^{2}-6 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**8 + 4*x**7 - 7*x**6 + 9*x**5 - 15*x**4 + 19*x**3 - 15*x**2 + 6*x - 1)/((x - 1)**4*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^4*F(x)+x^8-4*x^7+7*x^6-9*x^5+15*x^4-19*x^3+15*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-4 x^{7}+7 x^{6}-9 x^{5}+15 x^{4}-19 x^{3}+15 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/6900*(46*((I-4/23*23^(1/2))*3^(1/2)-12/23*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+9200-115*2^(2/3)*((I-1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/6900*(115*2^(2/3)*((I+1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200-46*((I+4/23*23^(1/2))*3^(1/2)-12/23*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/6900*((-10*2^(2/3)*23^(1/2)*3^(1/2)-230*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200+(16*23^(1/2)*3^(1/2)*2^(1/3)-92*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1/6*n^3-5/6*n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(46 \left(\left(\mathrm{I}-\frac{4 \sqrt{23}}{23}\right) \sqrt{3}-\frac{12 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+9200-115 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{6900}\\+\\\frac{\left(115 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200-46 \left(\left(\mathrm{I}+\frac{4 \sqrt{23}}{23}\right) \sqrt{3}-\frac{12 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{6900}\\+\\\frac{\left(\left(-10 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-230 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+\left(16 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}-92 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{6900}\\-\frac{n^{3}}{6}-\frac{5 n}{6}-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) = 54
a(6) = 144
a(7) = 368
a(8) = 911
a(n+3) = a(n)-2*a(n+1)+3*a(n+2)+1/6*(n+1)*(n^2+2*n+6), n >= 9
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) = 54
a \! \left(6\right) = 144
a \! \left(7\right) = 368
a \! \left(8\right) = 911
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{\left(n +1\right) \left(n^{2}+2 n +6\right)}{6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16136/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[39,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[26,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[7,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = 2*F[19,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[23,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[7,x]
F[38,x] = F[34,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[56,x]
F[41,x] = F[2,x]+F[42,x]
F[42,x] = F[19,x]+F[43,x]+F[55,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[4,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[7,x]
F[48,x] = F[42,x]+F[49,x]
F[49,x] = 2*F[19,x]+F[50,x]+F[54,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[46,x]
F[53,x] = F[49,x]
F[54,x] = F[34,x]*F[4,x]
F[55,x] = F[2,x]*F[4,x]
F[56,x] = F[34,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \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_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{55}\! \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_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{46}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{34}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_39(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_26(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_7(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), 2*F_19(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_7(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_56(x))
Eq(F_41(x), F_2(x) + F_42(x))
Eq(F_42(x), F_19(x) + F_43(x) + F_55(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_4(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_7(x))
Eq(F_48(x), F_42(x) + F_49(x))
Eq(F_49(x), 2*F_19(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_46(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_34(x)*F_4(x))
Eq(F_55(x), F_2(x)*F_4(x))
Eq(F_56(x), F_34(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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