0132_0231_1320_2013_3201
Counting sequence:
1, 1, 2, 6, 19, 49, 109, 221, 424, 790, 1456, 2688, 5009, 9455, 18087, 35015, 68446, 134780, 266802, 530066, 1055663, 2105757, 4204657, 8400961, 16791844, 33571634, 67128964, 134241076, 268462429, 536901915, 1073777291, 2147524043, 4295013114, 8589986360, 17179927462, 34359803750, 68719549851, 137439034985, 274877997557, 549755914341, 1099511738848, 2199023378062, 4398046645912, 8796093170216, 17592186206569, 35184372266119, 70368744371119, 140737488566031, 281474976939734, 562949953669940, 1125899907112026, 2251799813976698, 4503599627685319, 9007199255080565, 18014398509847737, 36028797019357385, 72057594038350556, 144115188076309290, 288230376152197612, 576460752303943516, 1152921504607402933, 2305843009214287667, 4611686018428021267, 9223372036855450771, 18446744073710270194, 36893488147419867504, 73786976294839018574, 147573952589677275086, 295147905179353740339, 590295810358706620865, 1180591620717412329661, 2361183241434823692653, 4722366482869646361624, 9444732965739291640070, 18889465931478582134912, 37778931862957163059920, 75557863725914324842561, 151115727451828648337695, 302231454903657295254967, 604462909807314589013591, 1208925819614629176451918, 2417851639229258351246572, 4835703278458516700750722, 9671406556917033399670626, 19342813113834066797418719, 38685626227668133592819789, 77371252455336267183523329, 154742504910672534364828241, 309485009821345068727332244, 618970019642690137452230690, 1237940039285380274901914196, 2475880078570760549801163908, 4951760157141521099599542029, 9903520314283042199196172875, 19807040628566084398389304987, 39614081257132168796775435355, 79228162514264337593547557866, 158456325028528675187091660200, 316912650057057350374179717622, 633825300114114700748355680566, 1267650600228229401496707449803
Generating function in Maple syntax:
-(2*x^7-7*x^6+10*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}-7 x^{6}+10 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 + 7*x**6 - 10*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^7-7*x^6+10*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}-7 x^{6}+10 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),1,2+1/12*n^3-13/24*n^2-19/12*n+1/24*n^4+2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2+\frac{n^{3}}{12}-\frac{13 n^{2}}{24}-\frac{19 n}{12}+\frac{n^{4}}{24}+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) = 49
a(6) = 109
a(7) = 221
a(n+1) = -1/24*n^4+1/12*n^3+25/24*n^2+2*a(n)+11/12*n-4, 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) = 49
a \! \left(6\right) = 109
a \! \left(7\right) = 221
a \! \left(n +1\right) = -\frac{n^{4}}{24}+\frac{n^{3}}{12}+\frac{25 n^{2}}{24}+2 a \! \left(n \right)+\frac{11 n}{12}-4, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19145/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[29,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[24,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[11,x]+F[21,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[11,x]
F[28,x] = F[21,x]
F[29,x] = F[30,x]+F[46,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[14,x]
F[34,x] = F[18,x]+F[35,x]
F[35,x] = F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[14,x]+F[35,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[18,x]+F[42,x]
F[42,x] = F[36,x]+F[43,x]+F[45,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]+F[42,x]
F[45,x] = F[4,x]*F[41,x]
F[46,x] = F[36,x]+F[47,x]+F[70,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[54,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[14,x]
F[54,x] = F[55,x]+F[64,x]
F[55,x] = F[36,x]+F[56,x]+F[58,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[55,x]+F[7,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[41,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[14,x]+F[61,x]
F[64,x] = 2*F[36,x]+F[65,x]+F[67,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[50,x]+F[64,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[61,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[85,x]
F[72,x] = F[30,x]+F[73,x]
F[73,x] = F[36,x]+F[74,x]+F[84,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[79,x]
F[76,x] = F[11,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[4,x]*F[53,x]
F[79,x] = F[42,x]+F[80,x]
F[80,x] = 2*F[36,x]+F[81,x]+F[83,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[77,x]+F[80,x]
F[83,x] = F[4,x]*F[69,x]
F[84,x] = F[4,x]*F[72,x]
F[85,x] = F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[89,x]
F[88,x] = F[14,x]
F[89,x] = F[61,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{29}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \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_{20}\! \left(x \right) F_{4}\! \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_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{46}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{36}\! \left(x \right)+F_{47}\! \left(x \right)+F_{70}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{64}\! \left(x \right)
F_{55}\! \left(x \right) = F_{36}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)+F_{7}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{41}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{50}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{61}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{85}\! \left(x \right)
F_{72}\! \left(x \right) = F_{30}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{36}\! \left(x \right)+F_{74}\! \left(x \right)+F_{84}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{11}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{79}\! \left(x \right) = F_{42}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{77}\! \left(x \right)+F_{80}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{14}\! \left(x \right)
F_{89}\! \left(x \right) = F_{61}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_29(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_11(x) + F_21(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_11(x))
Eq(F_28(x), F_21(x))
Eq(F_29(x), F_30(x) + F_46(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_14(x))
Eq(F_34(x), F_18(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_14(x) + F_35(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_18(x) + F_42(x))
Eq(F_42(x), F_36(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x) + F_42(x))
Eq(F_45(x), F_4(x)*F_41(x))
Eq(F_46(x), F_36(x) + F_47(x) + F_70(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_54(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_14(x))
Eq(F_54(x), F_55(x) + F_64(x))
Eq(F_55(x), F_36(x) + F_56(x) + F_58(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_55(x) + F_7(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_41(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_14(x) + F_61(x))
Eq(F_64(x), 2*F_36(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_50(x) + F_64(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_61(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_85(x))
Eq(F_72(x), F_30(x) + F_73(x))
Eq(F_73(x), F_36(x) + F_74(x) + F_84(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_79(x))
Eq(F_76(x), F_11(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_4(x)*F_53(x))
Eq(F_79(x), F_42(x) + F_80(x))
Eq(F_80(x), 2*F_36(x) + F_81(x) + F_83(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_77(x) + F_80(x))
Eq(F_83(x), F_4(x)*F_69(x))
Eq(F_84(x), F_4(x)*F_72(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_89(x))
Eq(F_88(x), F_14(x))
Eq(F_89(x), F_61(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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