0132_0321_1302_2301_2310
Counting sequence:
1, 1, 2, 6, 19, 54, 141, 348, 827, 1914, 4345, 9720, 21495, 47094, 102389, 221172, 475123, 1015794, 2162673, 4587504, 9699311, 20447214, 42991597, 90177516, 188743659, 394264554, 822083561, 1711276008, 3556769767, 7381975014, 15300820965, 31675383780, 65498251235, 135291469794, 279172874209, 575525617632, 1185410973663, 2439541424094, 5016521801693, 10307921510364, 21165598834651, 43430709297114, 89060441849817, 182518930210776, 373833953443799, 765260092932054, 1565704557952981, 3201777860083668, 6544293208522707, 13370061393756114, 27303072740933585, 55732045388709840, 113715890591104975, 231935380809580494, 472877960873902029, 963770320257286092, 1963569437533536203, 3999196469105000394, 8142508126285856713, 16573246628723425224, 33722954009750273991, 68598829524107395014, 139503502057428484037, 283618690133284356036, 576460752303423487939, 1171368248680556527554, 2379629985508532158401, 4833046947311902523328, 9813667847213481459647, 19922483599606315745214, 40435263009571337142205, 82051117639860085587900, 166463418521154993782715, 337649203525179632779194, 684743140016098555985849, 1388375745963675692826552, 2814530423790308547362743, 5704618711306531418144694, 11560353150064891483127733, 23422937755033440259932084, 47450338419874195107217331, 96109602659363019389140914, 194637056957955297127694257, 394109817194369110954213296, 797891040945655255306076079, 1615124895005144577407451054, 3268935416237957288405499821, 6615242084931250843992194988, 13385226674773174222346780587, 27079938359367693513418342314, 54778846738378077164286246825, 110795633516041534603471617960, 224067147110653829756741484455, 453086054378449180613079465894, 916075629071181403425351925669, 1851958298770928891249089839012, 3743530678798989951294951653283, 7566289520112244240183447256994, 15291035365253017155553982414753, 30898983380563091661482140630944, 62431792061240298023712632864671
Generating function in Maple syntax:
-(2*x^5-x^4+5*x^3-9*x^2+5*x-1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{5}-x^{4}+5 x^{3}-9 x^{2}+5 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**5 + x**4 - 5*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^2*F(x)+2*x^5-x^4+5*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+2 x^{5}-x^{4}+5 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3-n+2^(-1+n)*n-3/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3-n +2^{-1+n} n -\frac{3 \,2^{n}}{4} & \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(n+2) = -4*a(n)+4*a(n+1)-n+5, n >= 6
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(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-n +5, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19540/
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[23,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[12,x]+F[21,x]
F[21,x] = F[15,x]+F[19,x]+F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[24,x]+F[51,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[20,x]+F[9,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[1,x]+F[32,x]
F[35,x] = F[15,x]+F[36,x]+F[38,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[28,x]+F[35,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = F[15,x]+F[42,x]+F[44,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[10,x]+F[41,x]
F[44,x] = F[12,x]*F[40,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[32,x]
F[48,x] = 2*F[15,x]+F[49,x]+F[50,x]
F[49,x] = F[12,x]*F[41,x]
F[50,x] = F[12,x]*F[45,x]
F[51,x] = F[15,x]+F[52,x]+F[67,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[63,x]
F[54,x] = F[55,x]+F[6,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[59,x]+F[60,x]
F[59,x] = F[22,x]
F[60,x] = 2*F[15,x]+F[61,x]+F[62,x]
F[61,x] = F[12,x]*F[18,x]
F[62,x] = F[12,x]*F[58,x]
F[63,x] = F[64,x]+F[76,x]
F[64,x] = F[15,x]+F[65,x]+F[67,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[6,x]+F[64,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[40,x]+F[69,x]
F[69,x] = F[41,x]+F[70,x]
F[70,x] = F[15,x]+F[50,x]+F[71,x]+F[74,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[18,x]+F[73,x]
F[73,x] = 2*F[15,x]+F[71,x]+F[74,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[41,x]+F[73,x]
F[76,x] = 2*F[15,x]+F[77,x]+F[79,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[55,x]+F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[75,x]+F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[49,x]
F[83,x] = 3*F[15,x]+F[84,x]+F[85,x]
F[84,x] = F[12,x]*F[73,x]
F[85,x] = F[12,x]*F[81,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_{23}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{51}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{67}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{63}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{6}\! \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_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{76}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{6}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{40}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{41}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{41}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{55}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{75}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{49}\! \left(x \right)
F_{83}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \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_23(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_12(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_19(x) + F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_24(x) + F_51(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_20(x) + F_9(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_32(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_28(x) + F_35(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_10(x) + F_41(x))
Eq(F_44(x), F_12(x)*F_40(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_32(x))
Eq(F_48(x), 2*F_15(x) + F_49(x) + F_50(x))
Eq(F_49(x), F_12(x)*F_41(x))
Eq(F_50(x), F_12(x)*F_45(x))
Eq(F_51(x), F_15(x) + F_52(x) + F_67(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_63(x))
Eq(F_54(x), F_55(x) + F_6(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_59(x) + F_60(x))
Eq(F_59(x), F_22(x))
Eq(F_60(x), 2*F_15(x) + F_61(x) + F_62(x))
Eq(F_61(x), F_12(x)*F_18(x))
Eq(F_62(x), F_12(x)*F_58(x))
Eq(F_63(x), F_64(x) + F_76(x))
Eq(F_64(x), F_15(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_6(x) + F_64(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_40(x) + F_69(x))
Eq(F_69(x), F_41(x) + F_70(x))
Eq(F_70(x), F_15(x) + F_50(x) + F_71(x) + F_74(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_18(x) + F_73(x))
Eq(F_73(x), 2*F_15(x) + F_71(x) + F_74(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_41(x) + F_73(x))
Eq(F_76(x), 2*F_15(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_55(x) + F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_75(x) + F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_49(x))
Eq(F_83(x), 3*F_15(x) + F_84(x) + F_85(x))
Eq(F_84(x), F_12(x)*F_73(x))
Eq(F_85(x), F_12(x)*F_81(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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