0132_0213_1203_2301_2310
Counting sequence:
1, 1, 2, 6, 19, 53, 129, 282, 571, 1099, 2053, 3784, 6967, 12921, 24249, 46118, 88819, 172887, 339341, 670164, 1329263, 2644389, 5270977, 10519826, 21012459, 41991843, 83943829, 167840032, 335623591, 671180689, 1342283593, 2684476734, 5368848867, 10737577391, 21475016989, 42949876908, 85899575519, 171798949437, 343597671761, 687195088554, 1374389891803, 2748779465339, 5497558576677, 10995116760696, 21990233086999, 43980465694633, 87960930861529, 175921861143382, 351843721651411, 703687442607879, 1407374884457133, 2814749768087684, 5629499535276367, 11258999069576661, 22517998138095329, 45035996275045698, 90071992548854219, 180143985096373587, 360287970191308981, 720575940381070544, 1441151880760478343, 2882303761519172289, 5764607523036431977, 11529215046070816366, 23058430092139443139, 46116860184276547423, 92233720368550599229, 184467440737098538332, 368934881474194244031, 737869762948385474669, 1475739525896767746673, 2951479051793532092634, 5902958103587060577467, 11805916207174117330731, 23611832414348230611269, 47223664828696456936488, 94447329657392909340919, 188894659314785813893337, 377789318629571622731001, 755578637259143240128134, 1511157274518286474632883, 3022314549036572943341239, 6044629098073145880444877, 12089258196146291754326836, 24178516392292583501752879, 48357032784585166996254213, 96714065569170333984892929, 193428131138340667961792882, 386856262276681335915201451, 773712524553362671821613059, 1547425049106725343634016213, 3094850098213450687258387584, 6189700196426901374506680167, 12379400392853802749002799601, 24758800785707605497994556809, 49517601571415210995977573278, 99035203142830421991943091619, 198070406285660843983873596687, 396140812571321687967734057821, 792281625142643375935454413324, 1584563250285286751870894539423
Generating function in Maple syntax:
-(x^7+x^6+6*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{x^{7}+x^{6}+6 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**7 - x**6 - 6*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+x^7+x^6+6*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{7}+x^{6}+6 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3+5/4*2^n+10/3*n^2-19/3*n-7/6*n^3+1/6*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3+\frac{5 \,2^{n}}{4}+\frac{10 n^{2}}{3}-\frac{19 n}{3}-\frac{7 n^{3}}{6}+\frac{n^{4}}{6} & \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) = 53
a(6) = 129
a(7) = 282
a(n+1) = -1/6*n^4+11/6*n^3-35/6*n^2+2*a(n)+61/6*n-7, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 129
a \! \left(7\right) = 282
a \! \left(n +1\right) = -\frac{n^{4}}{6}+\frac{11 n^{3}}{6}-\frac{35 n^{2}}{6}+2 a \! \left(n \right)+\frac{61 n}{6}-7, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18429/
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[24,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[22,x]
F[24,x] = F[25,x]+F[47,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[14,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[14,x]+F[30,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[18,x]+F[37,x]
F[37,x] = F[31,x]+F[38,x]+F[40,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[10,x]+F[37,x]
F[40,x] = F[12,x]*F[36,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[18,x]+F[42,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[41,x]
F[47,x] = F[31,x]+F[48,x]+F[66,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[62,x]
F[50,x] = F[51,x]+F[6,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[17,x]+F[54,x]
F[54,x] = F[55,x]+F[61,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[17,x]+F[58,x]
F[58,x] = F[42,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[58,x]
F[61,x] = F[60,x]
F[62,x] = F[63,x]+F[72,x]
F[63,x] = F[31,x]+F[64,x]+F[66,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[6,x]+F[63,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[36,x]+F[68,x]
F[68,x] = F[30,x]+F[69,x]
F[69,x] = 2*F[31,x]+F[46,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[23,x]+F[69,x]
F[72,x] = 2*F[31,x]+F[73,x]+F[75,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[51,x]+F[72,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[41,x]+F[77,x]
F[77,x] = F[78,x]+F[87,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[41,x]+F[81,x]
F[81,x] = F[82,x]+F[85,x]
F[82,x] = F[83,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[42,x]+F[82,x]
F[85,x] = F[86,x]
F[86,x] = F[12,x]*F[81,x]
F[87,x] = F[86,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_{24}\! \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_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{48}\! \left(x \right)+F_{66}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{62}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{6}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{17}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{42}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{60}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{72}\! \left(x \right)
F_{63}\! \left(x \right) = F_{31}\! \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_{6}\! \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_{36}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{30}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{46}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{23}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{51}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{41}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{87}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{41}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{85}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{42}\! \left(x \right)+F_{82}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{87}\! \left(x \right) = F_{86}\! \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_24(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_22(x))
Eq(F_24(x), F_25(x) + F_47(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_14(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_14(x) + F_30(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_18(x) + F_37(x))
Eq(F_37(x), F_31(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_10(x) + F_37(x))
Eq(F_40(x), F_12(x)*F_36(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_18(x) + F_42(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_41(x))
Eq(F_47(x), F_31(x) + F_48(x) + F_66(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_62(x))
Eq(F_50(x), F_51(x) + F_6(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_17(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_61(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_17(x) + F_58(x))
Eq(F_58(x), F_42(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_58(x))
Eq(F_61(x), F_60(x))
Eq(F_62(x), F_63(x) + F_72(x))
Eq(F_63(x), F_31(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_6(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_36(x) + F_68(x))
Eq(F_68(x), F_30(x) + F_69(x))
Eq(F_69(x), 2*F_31(x) + F_46(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_23(x) + F_69(x))
Eq(F_72(x), 2*F_31(x) + F_73(x) + F_75(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_51(x) + F_72(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_41(x) + F_77(x))
Eq(F_77(x), F_78(x) + F_87(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_41(x) + F_81(x))
Eq(F_81(x), F_82(x) + F_85(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_42(x) + F_82(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_12(x)*F_81(x))
Eq(F_87(x), F_86(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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