0132_0213_1203_2310_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 132, 304, 665, 1407, 2914, 5954, 12063, 24313, 48848, 97956, 196213, 392771, 785934, 1572310, 3145115, 6290781, 12582172, 25165016, 50330769, 100662343, 201325562, 402652074, 805305175, 1610611457, 3221224104, 6442449484, 12884900333, 25769802123, 51539605798, 103079213246, 206158428243, 412316858341, 824633718644, 1649267439360, 3298534880905, 6597069764111, 13194139530642, 26388279063826, 52776558130319, 105553116263433, 211106232529792, 422212465062644, 844424930128485, 1688849860260307, 3377699720524094, 6755399441051814, 13510798882107403, 27021597764218733, 54043195528441548, 108086391056887336, 216172782113779073, 432345564227562711, 864691128455130154, 1729382256910265210, 3458764513820535495, 6917529027641076241, 13835058055282157912, 27670116110564321436, 55340232221128648669, 110680464442257303323, 221360928884514612822, 442721857769029232014, 885443715538058470595, 1770887431076116947957, 3541774862152233902884, 7083549724304467812944, 14167099448608935633273, 28334198897217871274143, 56668397794435742556098, 113336795588871485120226, 226673591177742970248703, 453347182355485940505881, 906694364710971881020464, 1813388729421943762049860, 3626777458843887524108885, 7253554917687775048227171, 14507109835375550096463982, 29014219670751100192937846, 58028439341502200385885819, 116056878683004400771782013, 232113757366008801543574652, 464227514732017603087160184, 928455029464035206174331505, 1856910058928070412348674407, 3713820117856140824697360474, 7427640235712281649394732874, 14855280471424563298789477943, 29710560942849126597578968353, 59421121885698253195157949448, 118842243771396506390315911916, 237684487542793012780631837133, 475368975085586025561263687851, 950737950171172051122527389574, 1901475900342344102245054793310, 3802951800684688204490109601075
Generating function in Maple syntax:
(2*x^4-2*x^3+6*x^2-4*x+1)/(2*x-1)/(-1+x)^3
Generating function in latex syntax:
\frac{2 x^{4}-2 x^{3}+6 x^{2}-4 x +1}{\left(2 x -1\right) \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(2*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^3*F(x)-2*x^4+2*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{3} F \! \left(x \right)-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,3*2^n-3/2*n^2-1/2*n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 3 \,2^{n}-\frac{3 n^{2}}{2}-\frac{n}{2}-3 & \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(n+1) = 2*a(n)+1/2*(3*n-2)*(n-1), n >= 5
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(n +1\right) = 2 a \! \left(n \right)+\frac{\left(3 n -2\right) \left(n -1\right)}{2}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18435/
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[12,x]*F[4,x]
F[4,x] = F[25,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[23,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[17,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[13,x]
F[25,x] = F[26,x]+F[42,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[1,x]+F[14,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[35,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[14,x]+F[31,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[18,x]+F[38,x]
F[38,x] = F[32,x]+F[39,x]+F[41,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[10,x]+F[38,x]
F[41,x] = F[12,x]*F[37,x]
F[42,x] = F[32,x]+F[43,x]+F[68,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[46,x]+F[6,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[13,x]
F[49,x] = F[50,x]+F[63,x]
F[50,x] = F[32,x]+F[51,x]+F[53,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[50,x]+F[6,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[37,x]+F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[14,x]+F[56,x]
F[59,x] = 2*F[32,x]+F[60,x]+F[62,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[23,x]+F[59,x]
F[62,x] = F[12,x]*F[55,x]
F[63,x] = 2*F[32,x]+F[64,x]+F[66,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[46,x]+F[63,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[55,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[80,x]
F[70,x] = F[26,x]+F[71,x]
F[71,x] = F[32,x]+F[72,x]+F[79,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[10,x]+F[23,x]
F[75,x] = F[38,x]+F[76,x]
F[76,x] = 2*F[32,x]+F[62,x]+F[77,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[23,x]+F[76,x]
F[79,x] = F[12,x]*F[70,x]
F[80,x] = F[81,x]+F[86,x]
F[81,x] = F[82,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[14,x]
F[85,x] = F[56,x]
F[86,x] = 2*F[32,x]+F[87,x]+F[91,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[23,x]
F[90,x] = F[59,x]
F[91,x] = F[12,x]*F[80,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{25}\! \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_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \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_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{42}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)+F_{68}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \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_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{63}\! \left(x \right)
F_{50}\! \left(x \right) = F_{32}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)+F_{6}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{37}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{23}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{46}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{55}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{80}\! \left(x \right)
F_{70}\! \left(x \right) = F_{26}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{32}\! \left(x \right)+F_{72}\! \left(x \right)+F_{79}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{75}\! \left(x \right) = F_{38}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{62}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{23}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{86}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{14}\! \left(x \right)
F_{85}\! \left(x \right) = F_{56}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{87}\! \left(x \right)+F_{91}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{23}\! \left(x \right)
F_{90}\! \left(x \right) = F_{59}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_25(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_23(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_17(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_13(x))
Eq(F_25(x), F_26(x) + F_42(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_1(x) + F_14(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_35(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_14(x) + F_31(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_18(x) + F_38(x))
Eq(F_38(x), F_32(x) + F_39(x) + F_41(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_10(x) + F_38(x))
Eq(F_41(x), F_12(x)*F_37(x))
Eq(F_42(x), F_32(x) + F_43(x) + F_68(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_46(x) + F_6(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_13(x))
Eq(F_49(x), F_50(x) + F_63(x))
Eq(F_50(x), F_32(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_50(x) + F_6(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_37(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_14(x) + F_56(x))
Eq(F_59(x), 2*F_32(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_23(x) + F_59(x))
Eq(F_62(x), F_12(x)*F_55(x))
Eq(F_63(x), 2*F_32(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_46(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_55(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_80(x))
Eq(F_70(x), F_26(x) + F_71(x))
Eq(F_71(x), F_32(x) + F_72(x) + F_79(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_10(x) + F_23(x))
Eq(F_75(x), F_38(x) + F_76(x))
Eq(F_76(x), 2*F_32(x) + F_62(x) + F_77(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_23(x) + F_76(x))
Eq(F_79(x), F_12(x)*F_70(x))
Eq(F_80(x), F_81(x) + F_86(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_14(x))
Eq(F_85(x), F_56(x))
Eq(F_86(x), 2*F_32(x) + F_87(x) + F_91(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_23(x))
Eq(F_90(x), F_59(x))
Eq(F_91(x), F_12(x)*F_80(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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