0132_1023_1230_1302_2130
Counting sequence:
1, 1, 2, 6, 19, 53, 138, 341, 812, 1881, 4266, 9515, 20940, 45583, 98338, 210567, 448060, 948385, 1998402, 4194841, 8776424, 18309813, 38104526, 79128151, 164006424, 339360803, 701155278, 1446727971, 2981521312, 6137861261, 12623128206, 25937281325, 53250201620, 109240848273, 223944314418, 458783095427, 939303390180, 1921994319415, 3930618074362, 8034251111871, 16414124144644, 33518810357113, 68417606754906, 139593287545921, 284699560531904, 580419826598253, 1182872305127318, 2409795353803951, 4907668008267312, 9991450468284491, 20335063234254246, 41374340635871835, 84156926630218600, 171130040958074213, 347891955758373798, 707046829456298837, 1436618122763729900, 2918282905554294345, 5926655384826275322, 12033483730541369051, 24427303170810112764, 49575260910775290463, 100591803166640171986, 204066123197365843191, 413897204590144411084, 839324201116244803153, 1701707781107312262258, 3449533982406831986281, 6991304249528611100312, 14167080154039791094821, 28703102113592566465886, 58144085363755842540295, 117763928931854551415880, 238479367583694103180403, 482861743614520860932478, 977529486060983971142547, 1978670940114568992508720, 4004565767483825553508925, 8103579229462197197614782, 16396053716558203908224381, 33169897732790944971897284, 67095375711149690470138113, 135701911333004746936099266, 274426141535319928374844211, 554896919247785525159762964, 1121883108289453432358522503, 2267944751969009348672381098, 4584246567837540848908401711, 9265207252197391198284881812, 18723842718960982490866953001, 37834541836779820467207871626, 76442796421681996014149314705, 154433018258381018318153774960, 311960887213774873898459280093, 630111475603766564302042033958, 1272602353203376553888850835807, 2569963509814720799381823980640, 5189444625489987387671508480155, 10477924461137557279062698918742, 21153919340031900348728602129611, 42703979511391603118461433939704
Generating function in Maple syntax:
-(x^10-3*x^9+4*x^8+8*x^7-7*x^6-13*x^5+7*x^4+16*x^3-18*x^2+7*x-1)/(x-1)^2/(2*x-1)^2/(x^2+x-1)^2
Generating function in latex syntax:
-\frac{x^{10}-3 x^{9}+4 x^{8}+8 x^{7}-7 x^{6}-13 x^{5}+7 x^{4}+16 x^{3}-18 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(-x**10 + 3*x**9 - 4*x**8 - 8*x**7 + 7*x**6 + 13*x**5 - 7*x**4 - 16*x**3 + 18*x**2 - 7*x + 1)/((x - 1)**2*(2*x - 1)**2*(x**2 + x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)^2*(x^2+x-1)^2*F(x)+x^10-3*x^9+4*x^8+8*x^7-7*x^6-13*x^5+7*x^4+16*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+x^{10}-3 x^{9}+4 x^{8}+8 x^{7}-7 x^{6}-13 x^{5}+7 x^{4}+16 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/400*((80*n+192)*5^(1/2)-240*n)*(-1/2*5^(1/2)-1/2)^(-n)+1/400*((-80*n-192)*5^(1/2)-240*n)*(1/2*5^(1/2)-1/2)^(-n)+1/400*(125*n+975)*2^n-n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(\left(80 n +192\right) \sqrt{5}-240 n \right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{400}+\\\frac{\left(\left(-80 n -192\right) \sqrt{5}-240 n \right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{400}+\frac{\left(125 n +975\right) 2^{n}}{400}-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) = 53
a(6) = 138
a(7) = 341
a(8) = 812
a(9) = 1881
a(10) = 4266
a(n+6) = -4*a(n)-4*a(n+1)+11*a(n+2)+2*a(n+3)-11*a(n+4)+6*a(n+5)-n+4, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 138
a \! \left(7\right) = 341
a \! \left(8\right) = 812
a \! \left(9\right) = 1881
a \! \left(10\right) = 4266
a \! \left(n +6\right) = -4 a \! \left(n \right)-4 a \! \left(n +1\right)+11 a \! \left(n +2\right)+2 a \! \left(n +3\right)-11 a \! \left(n +4\right)+6 a \! \left(n +5\right)-n +4, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19598/
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[7,x]
F[14,x] = F[15,x]+F[16,x]+F[26,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[16,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[16,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[15,x]+F[31,x]+F[75,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[41,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[1,x]+F[34,x]
F[37,x] = F[15,x]+F[16,x]+F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[38,x]
F[41,x] = F[42,x]+F[63,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[34,x]+F[46,x]
F[46,x] = F[16,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[58,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[22,x]+F[4,x]
F[56,x] = F[52,x]+F[57,x]
F[57,x] = F[49,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[52,x]
F[62,x] = F[59,x]
F[63,x] = 2*F[15,x]+F[49,x]+F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[70,x]
F[66,x] = F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[34,x]+F[67,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[67,x]
F[74,x] = F[71,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[79,x]
F[77,x] = F[2,x]+F[78,x]
F[78,x] = F[53,x]
F[79,x] = F[80,x]+F[85,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[84,x]
F[83,x] = F[2,x]
F[84,x] = F[80,x]
F[85,x] = F[59,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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{75}\! \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_{41}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{63}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{34}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{58}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{49}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{52}\! \left(x \right)
F_{62}\! \left(x \right) = F_{59}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{70}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{34}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)
F_{74}\! \left(x \right) = F_{71}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{2}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{53}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{85}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{84}\! \left(x \right)
F_{83}\! \left(x \right) = F_{2}\! \left(x \right)
F_{84}\! \left(x \right) = F_{80}\! \left(x \right)
F_{85}\! \left(x \right) = F_{59}\! \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_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_26(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_16(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_16(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_75(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_41(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_1(x) + F_34(x))
Eq(F_37(x), F_15(x) + F_16(x) + F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_38(x))
Eq(F_41(x), F_42(x) + F_63(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_34(x) + F_46(x))
Eq(F_46(x), F_16(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_58(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_22(x) + F_4(x))
Eq(F_56(x), F_52(x) + F_57(x))
Eq(F_57(x), F_49(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_52(x))
Eq(F_62(x), F_59(x))
Eq(F_63(x), 2*F_15(x) + F_49(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_70(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_34(x) + F_67(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_67(x))
Eq(F_74(x), F_71(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_79(x))
Eq(F_77(x), F_2(x) + F_78(x))
Eq(F_78(x), F_53(x))
Eq(F_79(x), F_80(x) + F_85(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_84(x))
Eq(F_83(x), F_2(x))
Eq(F_84(x), F_80(x))
Eq(F_85(x), F_59(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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