0132_0213_0231_2130_2301
Counting sequence:
1, 1, 2, 6, 19, 52, 122, 256, 499, 930, 1694, 3066, 5579, 10280, 19242, 36588, 70539, 137510, 270302, 534486, 1061171, 2112540, 4212922, 8410936, 16803779, 33585802, 67145662, 134260626, 268485179, 536928240, 1073807594, 2147558756, 4295052699, 8590031310, 17179978302, 34359861038, 68719614179, 137439106980, 274878077882, 549756003696, 1099511837971, 2199023487730, 4398046766942, 8796093303466, 17592186352939, 35184372426552, 70368744546602, 140737488757596, 281474977148459, 562949953896950, 1125899907358494, 2251799814243846, 4503599627974419, 9007199255392940, 18014398510184762, 36028797019720488, 72057594038741219, 144115188076729050, 288230376152648062, 576460752304426306, 1152921504607919771, 2305843009214840320, 4611686018428611562, 9223372036856080596, 18446744073710941499, 36893488147420582302, 73786976294839778942, 147573952589678083166, 295147905179354598339, 590295810358707531060, 1180591620717413294394, 2361183241434824714336, 4722366482869647442739, 9444732965739292783170, 18889465931478583342622, 37778931862957164334938, 75557863725914326187659, 151115727451828649755720, 302231454903657296748842, 604462909807314590586316, 1208925819614629178106571, 2417851639229258352986310, 4835703278458516702578782, 9671406556917033401590326, 19342813113834066799433459, 38685626227668133594933052, 77371252455336267185738682, 154742504910672534367149336, 309485009821345068729762819, 618970019642690137454774570, 1237940039285380274904575294, 2475880078570760549803946226, 4951760157141521099602449659, 9903520314283042199199210000, 19807040628566084398392475882, 39614081257132168796778744388, 79228162514264337593551009499, 158456325028528675187095258990, 316912650057057350374183468222, 633825300114114700748359587726, 1267650600228229401496711518371
Generating function in Maple syntax:
(2*x^7-x^6-7*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{2 x^{7}-x^{6}-7 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:
(2*x**7 - x**6 - 7*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)+2*x^7-x^6-7*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)+2 x^{7}-x^{6}-7 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,-5-25/12*n^2+5*n+1/12*n^4+2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -5-\frac{25 n^{2}}{12}+5 n +\frac{n^{4}}{12}+2^{n} & \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) = 52
a(6) = 122
a(7) = 256
a(n+1) = -1/12*n^4+1/3*n^3+31/12*n^2+2*a(n)-53/6*n+8, 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) = 52
a \! \left(6\right) = 122
a \! \left(7\right) = 256
a \! \left(n +1\right) = -\frac{n^{4}}{12}+\frac{n^{3}}{3}+\frac{31 n^{2}}{12}+2 a \! \left(n \right)-\frac{53 n}{6}+8, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18207/
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[19,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[33,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[5,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[13,x]+F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[14,x]+F[28,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[27,x]
F[33,x] = F[34,x]+F[35,x]+F[78,x]
F[34,x] = 0
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[47,x]
F[37,x] = F[38,x]+F[6,x]
F[38,x] = F[25,x]+F[34,x]+F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[41,x]
F[44,x] = F[17,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[44,x]
F[47,x] = F[33,x]+F[48,x]
F[48,x] = 2*F[34,x]+F[49,x]+F[65,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[59,x]
F[51,x] = F[52,x]+F[57,x]
F[52,x] = F[34,x]+F[53,x]+F[55,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[10,x]+F[52,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[14,x]+F[52,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[51,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[28,x]+F[60,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[59,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[62,x]+F[67,x]
F[67,x] = F[68,x]+F[71,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[28,x]+F[68,x]
F[71,x] = F[72,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[74,x]+F[77,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[28,x]+F[74,x]
F[77,x] = F[72,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[56,x]+F[80,x]
F[80,x] = F[81,x]+F[86,x]
F[81,x] = F[34,x]+F[82,x]+F[84,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[14,x]+F[28,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[14,x]+F[81,x]
F[86,x] = 2*F[34,x]+F[61,x]+F[87,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[17,x]+F[60,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_{19}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{33}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{5}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \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_{13}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \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_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{78}\! \left(x \right)
F_{34}\! \left(x \right) = 0
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_{47}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{6}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{33}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{49}\! \left(x \right)+F_{65}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{59}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right)+F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{28}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{28}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \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_{77}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{28}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{72}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{56}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{86}\! \left(x \right)
F_{81}\! \left(x \right) = F_{34}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{14}\! \left(x \right)+F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{61}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{17}\! \left(x \right)+F_{60}\! \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_19(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_33(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_5(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_13(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_14(x) + F_28(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_27(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_78(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_47(x))
Eq(F_37(x), F_38(x) + F_6(x))
Eq(F_38(x), F_25(x) + F_34(x) + F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_41(x))
Eq(F_44(x), F_17(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_44(x))
Eq(F_47(x), F_33(x) + F_48(x))
Eq(F_48(x), 2*F_34(x) + F_49(x) + F_65(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_59(x))
Eq(F_51(x), F_52(x) + F_57(x))
Eq(F_52(x), F_34(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_10(x) + F_52(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_14(x) + F_52(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_51(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_28(x) + F_60(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_59(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_62(x) + F_67(x))
Eq(F_67(x), F_68(x) + F_71(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_28(x) + F_68(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_77(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_28(x) + F_74(x))
Eq(F_77(x), F_72(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_56(x) + F_80(x))
Eq(F_80(x), F_81(x) + F_86(x))
Eq(F_81(x), F_34(x) + F_82(x) + F_84(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_14(x) + F_28(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_14(x) + F_81(x))
Eq(F_86(x), 2*F_34(x) + F_61(x) + F_87(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_17(x) + F_60(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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