0132_0213_0321_3120
Counting sequence:
1, 1, 2, 6, 20, 63, 182, 493, 1285, 3285, 8336, 21138, 53733, 137091, 351111, 902462, 2326889, 6015730, 15587785, 40467142, 105221234, 273948032, 713997167, 1862533111, 4862061230, 12699525258, 33186173122, 86754124783, 226858083470, 593367124325, 1552303746096, 4061597934403, 10628463499203, 27815471034031, 72800769699266, 190551404544272, 498780429444859, 1305639559884405, 3417829012506065, 8947211822409972, 23422500784594199, 61317610471699932, 160524833072277707, 420245618750849716, 1100188933435978840, 2880273902556971778, 7540536017211559337, 19741136236984560573, 51682468073462940780, 135305441150659999140, 354232166528656601556, 927387610366844891693, 2427923627697460080548, 6356368917501722860471, 16641153851410130325946, 43567032964033605169657, 114059823443500745867729, 298612189668489126719705, 781776241158808368433220, 2046715506987220937716702, 5358368190132627344387489, 14028384812012612857247607, 36726777598993926675520747, 96151930402916221914381946, 251728977869188096254816725, 659034930570593276618118998, 1725375666268639143922498077, 4517091768475732957368490738, 11825899030416005295767056590, 30960604086840429991391960980, 81055910721348090653908999091, 212207122985902477626497932883, 555565447906182660948234985018, 1454489199777144237484155422286, 3807902108923951705677423337526, 9969217040811522567177088359867, 26099748838783056129676967692874, 68330029121360159606630422484505, 178890337807497717294028226004500, 468340982846644115551603528366207, 1226132607785677944050123744330479, 3210056834541318482251536153020307, 8404037883749019306558192966096662, 22002056792224991590226801955815676, 57602132443359996859922416736658135, 150804340437514155961526226751718793, 394810888666082933329398562165941717, 1033628325149699865357695540343992528, 2706074085951275698848823186666074275, 7084593931021302490285210208461018720, 18547707703708296663972011682742069951, 48558529173217546033105897060636382848, 127127879802017515993385151406549387988, 332825110204671866053307155905054044726, 871347450755052840359408823663250250381, 2281217241945358231371378949519075246777, 5972304274848289126369076533212937099544, 15635695582129091932807406145658065676680, 40934782470588248721881969852638729218280, 107168651827714371291867593225614677731737, 280571173008672685190521857282121646812544
Generating function in Maple syntax:
(x^10-6*x^9+3*x^8+48*x^7-144*x^6+224*x^5-220*x^4+137*x^3-52*x^2+11*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{x^{10}-6 x^{9}+3 x^{8}+48 x^{7}-144 x^{6}+224 x^{5}-220 x^{4}+137 x^{3}-52 x^{2}+11 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**10 - 6*x**9 + 3*x**8 + 48*x**7 - 144*x**6 + 224*x**5 - 220*x**4 + 137*x**3 - 52*x**2 + 11*x - 1)/((x - 1)**5*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2*(x-1)^5*F(x)+x^10-6*x^9+3*x^8+48*x^7-144*x^6+224*x^5-220*x^4+137*x^3-52*x^2+11*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+x^{10}-6 x^{9}+3 x^{8}+48 x^{7}-144 x^{6}+224 x^{5}-220 x^{4}+137 x^{3}-52 x^{2}+11 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/24*n^4-5/12*n^3+1/8*2^n*n+11/24*n^2+1/4*2^n-13/12*n-1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{n^{4}}{24}-\frac{5 n^{3}}{12}+\frac{2^{n} n}{8}+\frac{11 n^{2}}{24}+\frac{2^{n}}{4}-\frac{13 n}{12}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 182
a(7) = 493
a(8) = 1285
a(9) = 3285
a(10) = 8336
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1/24*(n-4)*(n^3-10*n^2-29*n+6), n >= 11
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) = 63
a \! \left(6\right) = 182
a \! \left(7\right) = 493
a \! \left(8\right) = 1285
a \! \left(9\right) = 3285
a \! \left(10\right) = 8336
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{\left(n -4\right) \left(n^{3}-10 n^{2}-29 n +6\right)}{24}, \quad n \geq 11
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21179/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[19,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[61,x]
F[4,x] = F[19,x]*F[5,x]
F[5,x] = F[1,x]+F[45,x]+F[46,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[19,x]*F[8,x]
F[8,x] = F[1,x]+F[30,x]+F[31,x]+F[9,x]
F[9,x] = F[10,x]*F[19,x]
F[10,x] = 2*F[29,x]+F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[16,x]*F[19,x]
F[13,x] = F[1,x]+F[14,x]+F[9,x]
F[14,x] = F[15,x]*F[19,x]
F[15,x] = F[16,x]+F[20,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[16,x]*F[19,x]
F[19,x] = x
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[19,x]*F[23,x]
F[23,x] = F[1,x]+F[22,x]
F[24,x] = F[25,x]+F[26,x]+F[28,x]
F[25,x] = 0
F[26,x] = F[19,x]*F[27,x]
F[27,x] = F[17,x]+F[24,x]
F[28,x] = F[19,x]*F[20,x]
F[29,x] = F[10,x]*F[19,x]
F[30,x] = F[19,x]*F[23,x]
F[31,x] = F[19,x]*F[32,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[16,x]+F[34,x]
F[34,x] = F[19,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[17,x]*F[19,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[19,x]*F[41,x]
F[41,x] = F[19,x]+F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[19,x]*F[44,x]
F[44,x] = F[35,x]+F[42,x]
F[45,x] = F[19,x]*F[5,x]
F[46,x] = F[19,x]*F[47,x]
F[47,x] = F[1,x]+F[46,x]+F[48,x]+F[51,x]+F[53,x]
F[48,x] = F[49,x]
F[49,x] = F[16,x]*F[19,x]*F[50,x]
F[50,x] = F[30,x]+F[8,x]
F[51,x] = F[19,x]*F[52,x]
F[52,x] = 2*F[51,x]+F[1,x]+F[6,x]
F[53,x] = F[19,x]*F[54,x]
F[54,x] = F[1,x]+F[53,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[16,x]*F[19,x]*F[57,x]
F[57,x] = F[30,x]+F[58,x]
F[58,x] = F[1,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[19,x]*F[23,x]*F[58,x]
F[61,x] = F[19,x]*F[62,x]
F[62,x] = F[1,x]+F[14,x]+F[61,x]+F[63,x]
F[63,x] = F[19,x]*F[64,x]
F[64,x] = F[1,x]+F[11,x]+F[65,x]+F[66,x]+F[68,x]
F[65,x] = F[19,x]*F[64,x]
F[66,x] = F[67,x]
F[67,x] = F[15,x]*F[19,x]*F[58,x]
F[68,x] = F[19,x]*F[69,x]
F[69,x] = F[1,x]+F[11,x]+F[29,x]+F[68,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[19,x]*F[23,x]*F[58,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_{19}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{61}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \left(x \right)
F_{10}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = x
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{19}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right) F_{20}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right) F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{19}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{19}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right) F_{5}\! \left(x \right)
F_{46}\! \left(x \right) = F_{19}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{30}\! \left(x \right)+F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{51}\! \left(x \right)+F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{53}\! \left(x \right) = F_{19}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{1}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{30}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{1}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{19}\! \left(x \right) F_{23}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{19}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{19}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{19}\! \left(x \right) F_{64}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{58}\! \left(x \right)
F_{68}\! \left(x \right) = F_{19}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{29}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{19}\! \left(x \right) F_{23}\! \left(x \right) F_{58}\! \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_19(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_61(x))
Eq(F_4(x), F_19(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_45(x) + F_46(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_16(x)*F_19(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_30(x) + F_31(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_19(x))
Eq(F_10(x), F_1(x) + F_11(x) + 2*F_29(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_16(x)*F_19(x))
Eq(F_13(x), F_1(x) + F_14(x) + F_9(x))
Eq(F_14(x), F_15(x)*F_19(x))
Eq(F_15(x), F_16(x) + F_20(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_16(x)*F_19(x))
Eq(F_19(x), x)
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_19(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_22(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_28(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_19(x)*F_27(x))
Eq(F_27(x), F_17(x) + F_24(x))
Eq(F_28(x), F_19(x)*F_20(x))
Eq(F_29(x), F_10(x)*F_19(x))
Eq(F_30(x), F_19(x)*F_23(x))
Eq(F_31(x), F_19(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_16(x) + F_34(x))
Eq(F_34(x), F_19(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_17(x)*F_19(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_19(x)*F_41(x))
Eq(F_41(x), F_19(x) + F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_19(x)*F_44(x))
Eq(F_44(x), F_35(x) + F_42(x))
Eq(F_45(x), F_19(x)*F_5(x))
Eq(F_46(x), F_19(x)*F_47(x))
Eq(F_47(x), F_1(x) + F_46(x) + F_48(x) + F_51(x) + F_53(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_16(x)*F_19(x)*F_50(x))
Eq(F_50(x), F_30(x) + F_8(x))
Eq(F_51(x), F_19(x)*F_52(x))
Eq(F_52(x), F_1(x) + 2*F_51(x) + F_6(x))
Eq(F_53(x), F_19(x)*F_54(x))
Eq(F_54(x), F_1(x) + F_53(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_16(x)*F_19(x)*F_57(x))
Eq(F_57(x), F_30(x) + F_58(x))
Eq(F_58(x), F_1(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_19(x)*F_23(x)*F_58(x))
Eq(F_61(x), F_19(x)*F_62(x))
Eq(F_62(x), F_1(x) + F_14(x) + F_61(x) + F_63(x))
Eq(F_63(x), F_19(x)*F_64(x))
Eq(F_64(x), F_1(x) + F_11(x) + F_65(x) + F_66(x) + F_68(x))
Eq(F_65(x), F_19(x)*F_64(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_15(x)*F_19(x)*F_58(x))
Eq(F_68(x), F_19(x)*F_69(x))
Eq(F_69(x), F_1(x) + F_11(x) + F_29(x) + F_68(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_19(x)*F_23(x)*F_58(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_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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