0231_1032_2103_3012
Counting sequence:
1, 1, 2, 6, 20, 58, 147, 358, 864, 2082, 5019, 12106, 29212, 70506, 170195, 410862, 991880, 2394578, 5780987, 13956498, 33693924, 81344282, 196382419, 474109046, 1144600432, 2763309826, 6671219995, 16105749722, 38882719340, 93871188298, 226625095827, 547121379838, 1320867855384, 3188857090482, 7698582036219, 18586021162786, 44870624361652, 108327269885946, 261525164133395, 631377598152582, 1524280360438400, 3679938319029218, 8884156998496667, 21448252316022378, 51780661630541244, 125009575577104682, 301799812784750419, 728609201146605326, 1759018215077960872, 4246645631302526866, 10252309477683014395, 24751264586668555442, 59754838651020125060, 144260941888708805338, 348276722428437735507, 840814386745584276118, 2029905495919606287504, 4900625378584796850882, 11831156253089199989019, 28562937884763196828666, 68957032022615593646092, 166477001929994384120586, 401911035882604361886995, 970299073695203107894302, 2342509183273010577675320, 5655317440241224263244658, 13653144063755459104164347, 32961605567752142471573058, 79576355199259744047310164, 192114315966271630566193082, 463804987131803005179696019, 1119724290229877640925584806, 2703253567591558287030865312, 6526231425412994214987315106, 15755716418417546717005495195, 38037664262248087648998305162, 91831044942913722015002105180, 221699754148075531679002515178, 535230553239064785373007135187, 1292160860626205102425016785198, 3119552274491474990223040705224, 7531265409609155082871098195282, 18182083093709785155965237095419, 43895431597028725394801572385746, 105972946287767235945568381866532, 255841324172563197285938336118426, 617655594632893630517445054102995, 1491152513438350458320828444324022, 3599960621509594547159101942750640, 8691073756457539552639032329824898, 20982108134424673652437166602400027, 50655290025306886857513365534624538, 122292688185038447367463897671648684, 295240666395383781592441160877921482, 712774020975806010552346219427491219, 1720788708346995802697133599732903486, 4154351437669797615946613418893297752, 10029491583686591034590360437519498546, 24213334605042979685127334293932294395, 58456160793772550404845029025384086882, 141125656192588080494817392344700467700
Generating function in Maple syntax:
(4*x^7+7*x^6-x^5-3*x^4-2*x^3-2*x^2+3*x-1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{4 x^{7}+7 x^{6}-x^{5}-3 x^{4}-2 x^{3}-2 x^{2}+3 x -1}{\left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(4*x**7 + 7*x**6 - x**5 - 3*x**4 - 2*x**3 - 2*x**2 + 3*x - 1)/((x - 1)**2*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+2*x-1)*(x-1)^2*F(x)+4*x^7+7*x^6-x^5-3*x^4-2*x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{7}+7 x^{6}-x^{5}-3 x^{4}-2 x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,9,n = 1,15,n = 2,7,n = 3,4,0)+1/8*(17*2^(1/2)+30)*(-1-2^(1/2))^(-n)+1/8*(-17*2^(1/2)+30)*(2^(1/2)-1)^(-n)-31/2+5/2*n
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}9 & n =0 \\ 15 & n =1 \\ 7 & n =2 \\ 4 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(17 \sqrt{2}+30\right) \left(-1-\sqrt{2}\right)^{-n}}{8}+\frac{\left(-17 \sqrt{2}+30\right) \left(\sqrt{2}-1\right)^{-n}}{8}-\frac{31}{2}+\frac{5 n}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 58
a(6) = 147
a(7) = 358
a(n+2) = a(n)+2*a(n+1)-5*n+31, n >= 8
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) = 20
a \! \left(5\right) = 58
a \! \left(6\right) = 147
a \! \left(7\right) = 358
a \! \left(n +2\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-5 n +31, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15881/
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[31,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[21,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[25,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[25,x]
F[31,x] = F[32,x]+F[49,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[14,x]+F[35,x]
F[35,x] = F[32,x]+F[36,x]
F[36,x] = F[37,x]+F[38,x]+F[48,x]
F[37,x] = 0
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[4,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[15,x]*F[4,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[45,x]
F[45,x] = F[32,x]*F[4,x]
F[46,x] = F[47,x]
F[47,x] = F[36,x]*F[4,x]
F[48,x] = F[35,x]*F[4,x]
F[49,x] = F[37,x]+F[50,x]+F[66,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[56,x]
F[52,x] = F[25,x]+F[53,x]
F[53,x] = F[29,x]+F[37,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[15,x]+F[28,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[32,x]+F[57,x]
F[60,x] = 2*F[37,x]+F[61,x]+F[63,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[57,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[36,x]+F[65,x]
F[65,x] = F[61,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[31,x]+F[68,x]
F[68,x] = F[69,x]+F[76,x]
F[69,x] = F[37,x]+F[38,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[35,x]+F[72,x]
F[72,x] = F[69,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[44,x]
F[76,x] = 2*F[37,x]+F[77,x]+F[83,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[81,x]
F[79,x] = F[80,x]
F[80,x] = F[25,x]*F[4,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[57,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]
F[85,x] = F[57,x]+F[65,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_{31}\! \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_{21}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \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_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{49}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)+F_{66}\! \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_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{29}\! \left(x \right)+F_{37}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{32}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{36}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{31}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{76}\! \left(x \right)
F_{69}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{35}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{44}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{77}\! \left(x \right)+F_{83}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{57}\! \left(x \right)+F_{65}\! \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_31(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_21(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_25(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_32(x) + F_49(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_14(x) + F_35(x))
Eq(F_35(x), F_32(x) + F_36(x))
Eq(F_36(x), F_37(x) + F_38(x) + F_48(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_4(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_15(x)*F_4(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_32(x)*F_4(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_36(x)*F_4(x))
Eq(F_48(x), F_35(x)*F_4(x))
Eq(F_49(x), F_37(x) + F_50(x) + F_66(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_56(x))
Eq(F_52(x), F_25(x) + F_53(x))
Eq(F_53(x), F_29(x) + F_37(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_15(x) + F_28(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_32(x) + F_57(x))
Eq(F_60(x), 2*F_37(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_57(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_36(x) + F_65(x))
Eq(F_65(x), F_61(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_31(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_76(x))
Eq(F_69(x), F_37(x) + F_38(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_35(x) + F_72(x))
Eq(F_72(x), F_69(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_44(x))
Eq(F_76(x), 2*F_37(x) + F_77(x) + F_83(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_81(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_25(x)*F_4(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_57(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_57(x) + F_65(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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