0123_0132_1203_2031_3102
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 371, 935, 2323, 5717, 13981, 34046, 82672, 200365, 484986, 1172907, 2834966, 6849588, 16545071, 39957423, 96488555, 232980881, 562525321, 1358152894, 3279027504, 7916525689, 19112593086, 46142543875, 111399027078, 268942776312, 649288104251, 1567524687671, 3784346707027, 9136233032045, 22056836928901, 53249945977982, 128556792130816, 310363632573733, 749284222858386, 1808932346204763, 4367149348762310, 10543231745138076, 25453613973941591, 61450461529333119, 148354540003822859, 358159546344505769, 864673640471576401, 2087506839873927550, 5039687340584442528, 12166881553994092657, 29373450501888918966, 70913782644039501811, 171201015929551784982, 413315814728994505440, 997832645752976091971, 2408981106826233419207, 5815794860362164956371, 14040570829098572087813, 33896936521064039913901, 81834443875279391453438, 197565824278180293140560, 476966092442250187592221, 1151498009179848348502506, 2779962110829724774632459, 6711422230884243467980214, 16202806572670935170840964, 39117035376343782840123071, 94436877325548893341796175, 227990790027749631044885483, 550418457381546609443446337, 1328827704791649365464827481, 3208073866966150309918029886, 7744975438726061470378865232, 18698024744421689705298666985, 45141024927574968820677083886, 108980074599580571740976626147, 263101174126750584636655012326, 635182422853105157742635118408, 1533466019832998789184298392971, 3702114462519164041901953515863, 8937694944871426067841300180115, 21577504352262176678228370243101, 52092703649396039119794951788821, 125762911651054675113959001310334, 303618526951506069239350593021664, 732999965554067913680438553455509, 1769618458059643676579643704646786, 4272236881673358146906920333565115, 10314092221406364630440094747307238, 24900421324486094947900914574525932, 60114934870378566726402339018235751, 145130291065243248140979812479220511, 350375517000865094948796598966776587, 845881325066973489719281865271096665, 2042138167134812158008503819357392801, 4930157659336597941038141848692628222, 11902453485808008259007783351297818176, 28735064630952614813278556730550179553, 69372582747713238458712740826215261286, 167480230126379092658076730576059701203
Generating function in Maple syntax:
(x^8+2*x^7-x^6-3*x^5-2*x^4+3*x^3+4*x^2-4*x+1)/(x^2+x-1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{8}+2 x^{7}-x^{6}-3 x^{5}-2 x^{4}+3 x^{3}+4 x^{2}-4 x +1}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**8 + 2*x**7 - x**6 - 3*x**5 - 2*x**4 + 3*x**3 + 4*x**2 - 4*x + 1)/((x - 1)**2*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^2+2*x-1)*(x-1)^2*F(x)+x^8+2*x^7-x^6-3*x^5-2*x^4+3*x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}+2 x^{7}-x^{6}-3 x^{5}-2 x^{4}+3 x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,3,Or(n = 1,n = 2),1,0)+1/2*n+2/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-2/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+5/8*(2^(1/2)-1)^(-n)*2^(1/2)-5/8*(-1-2^(1/2))^(-n)*2^(1/2)-(-1/2*5^(1/2)-1/2)^(-n)-(1/2*5^(1/2)-1/2)^(-n)
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}3 & n =0 \\ 1 & n =1\text{ or } n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{n}{2}+\frac{2 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\frac{2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{5 \left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{8}-\frac{5 \left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{8}-\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 144
a(7) = 371
a(8) = 935
a(n+1) = -1/3*a(n)+a(n+3)-1/3*a(n+4)-1/3+1/3*n, n >= 9
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(6\right) = 144
a \! \left(7\right) = 371
a \! \left(8\right) = 935
a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}-\frac{1}{3}+\frac{n}{3}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16446/
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[24,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[18,x]
F[24,x] = F[25,x]+F[39,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[30,x]*F[4,x]
F[39,x] = F[19,x]+F[40,x]+F[53,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[43,x]+F[7,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[22,x]+F[46,x]
F[46,x] = F[44,x]
F[47,x] = F[39,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[52,x]
F[51,x] = F[34,x]
F[52,x] = F[49,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[33,x]+F[55,x]
F[55,x] = F[56,x]+F[72,x]
F[56,x] = F[19,x]+F[57,x]+F[67,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[63,x]
F[59,x] = F[12,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[22,x]
F[63,x] = F[30,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[51,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[33,x]+F[69,x]
F[69,x] = F[56,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[56,x]
F[72,x] = 2*F[19,x]+F[71,x]+F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[51,x]+F[75,x]
F[75,x] = F[72,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_{24}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \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_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \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_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \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_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \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_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{53}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{39}\! \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_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{33}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{72}\! \left(x \right)
F_{56}\! \left(x \right) = F_{19}\! \left(x \right)+F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{63}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{22}\! \left(x \right)
F_{63}\! \left(x \right) = F_{30}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{51}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{33}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{56}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{51}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{72}\! \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_24(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_18(x))
Eq(F_24(x), F_25(x) + F_39(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_12(x))
Eq(F_29(x), F_25(x) + F_30(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_36(x))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_30(x)*F_4(x))
Eq(F_39(x), F_19(x) + F_40(x) + F_53(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_43(x) + F_7(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_22(x) + F_46(x))
Eq(F_46(x), F_44(x))
Eq(F_47(x), F_39(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_52(x))
Eq(F_51(x), F_34(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_33(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_72(x))
Eq(F_56(x), F_19(x) + F_57(x) + F_67(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_63(x))
Eq(F_59(x), F_12(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_22(x))
Eq(F_63(x), F_30(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_51(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_33(x) + F_69(x))
Eq(F_69(x), F_56(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_56(x))
Eq(F_72(x), 2*F_19(x) + F_71(x) + F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_51(x) + F_75(x))
Eq(F_75(x), F_72(x))
Pack JSON:
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Specification JSON:
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