0231_1032_2130_3021
Counting sequence:
1, 1, 2, 6, 20, 62, 180, 503, 1377, 3726, 10007, 26730, 71095, 188447, 498132, 1313866, 3459498, 9096954, 23896354, 62722515, 164532535, 431398426, 1130710197, 2962828006, 7761966605, 20331458667, 53249184610, 139449647318, 365166863632, 956185158406, 2503657043792, 6555322840255, 17163385214765, 44936980283222, 117651850597267, 308027161437746, 806446813579203, 2111347639031719, 5527664822985584, 14471784268870770, 37887962861525270, 99192654071509826, 259691098864622142, 679882841545601531, 1779961823818682115, 4660011426003454722, 12200090046377713265, 31940293897501759758, 83620862014871728537, 218922432884601765011, 573146718113910259902, 1500518284407082417630, 3928409261007243816060, 10284711750414462695022, 26925730493835771617452, 70492488738292106874951, 184551753735439058464633, 483162808496822087456734, 1264936743812621241805839, 3311647567056229713787450, 8670006245586444051237455, 22698371746163854743315951, 59425110145826624785523212, 155576960997159028826911706, 407305777457336480122562050, 1066340380598222448395510602, 2791715382784074938773479770, 7308805804647490515343988323, 19134702104945372902096645967, 50095300657762580780622314666, 131151200163490274619123073837, 343358300423004053435452506230, 898923702286113506404645693605, 2353412808796519707213307124347, 6161314728825812098104920833562, 16130531387125649552840745741606, 42230279451440602491895897181288, 110560307004975089785804107444374, 289450641639042530591430748500744, 757791618063268229440316784823055, 1983924212852993612633176899568965, 5193981021100175518266528501157926, 13598018851656458761781037778528907, 35600075536287052406305843183755906, 93202207762040401735595008471475035, 244006547759505559357396215628226807, 638817435535819089450427705208409000, 1672445758886637335221555033587499138, 4378519841201464168669573662735181422, 11463113764872497675697838488980329746, 30010821453725513868245286872930479222, 78569350596923013948680712267260556555, 205697230338281468017082230203750197275, 538522340420397270181136738893788162322, 1409869790927862102683469507577210661097, 3691087032373092558183554826037036685086, 9663391306211222612495761054932285247793, 25299086886300189360560860507556590895139, 66233869352768573631701084805331031245174, 173402521172163987859570922583623590593710, 453973694164040302597068740295913916185300
Generating function in Maple syntax:
(2*x^7-10*x^6+22*x^5-39*x^4+43*x^3-26*x^2+8*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{2 x^{7}-10 x^{6}+22 x^{5}-39 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(2*x**7 - 10*x**6 + 22*x**5 - 39*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^4*F(x)-2*x^7+10*x^6-22*x^5+39*x^4-43*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-2 x^{7}+10 x^{6}-22 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(3*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-3*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-n^2+11/6*n-2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-n^{2}+\frac{11 n}{6}\\-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) = 20
a(5) = 62
a(6) = 180
a(7) = 503
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1/6*(n-2)*(n^2-4*n-9), 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 180
a \! \left(7\right) = 503
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{\left(n -2\right) \left(n^{2}-4 n -9\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15885/
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[13,x]*F[4,x]
F[4,x] = F[26,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[16,x]+F[24,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[15,x]+F[16,x]+F[22,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[10,x]+F[21,x]
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[9,x]
F[26,x] = F[27,x]+F[42,x]
F[27,x] = F[28,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[30,x]+F[38,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[34,x]
F[38,x] = F[27,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[13,x]*F[41,x]
F[41,x] = F[38,x]
F[42,x] = F[15,x]+F[43,x]+F[60,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[45,x]+F[53,x]
F[45,x] = F[18,x]+F[46,x]
F[46,x] = F[15,x]+F[22,x]+F[47,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[45,x]+F[49,x]
F[49,x] = F[21,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[49,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[15,x]+F[43,x]+F[55,x]
F[55,x] = F[13,x]*F[56,x]
F[56,x] = F[27,x]+F[54,x]
F[57,x] = F[58,x]
F[58,x] = F[13,x]*F[59,x]
F[59,x] = F[53,x]
F[60,x] = F[13,x]*F[61,x]
F[61,x] = F[26,x]+F[62,x]
F[62,x] = F[63,x]+F[70,x]
F[63,x] = F[64,x]
F[64,x] = F[13,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[13,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[13,x]*F[69,x]
F[69,x] = F[66,x]
F[70,x] = 2*F[15,x]+F[71,x]+F[79,x]
F[71,x] = F[13,x]*F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[74,x]+F[76,x]
F[74,x] = F[75,x]
F[75,x] = F[13,x]*F[18,x]
F[76,x] = F[77,x]
F[77,x] = F[13,x]*F[78,x]
F[78,x] = F[73,x]
F[79,x] = F[13,x]*F[80,x]
F[80,x] = F[62,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{26}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \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_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{21}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{27}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{13}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{53}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{26}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{70}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{13}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{13}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{13}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{71}\! \left(x \right)+F_{79}\! \left(x \right)
F_{71}\! \left(x \right) = F_{13}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{13}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{73}\! \left(x \right)
F_{79}\! \left(x \right) = F_{13}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{62}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_26(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_16(x) + F_24(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(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_13(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_15(x) + F_16(x) + F_22(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_10(x) + F_21(x))
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_9(x))
Eq(F_26(x), F_27(x) + F_42(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_38(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_34(x))
Eq(F_38(x), F_27(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_13(x)*F_41(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_60(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_53(x))
Eq(F_45(x), F_18(x) + F_46(x))
Eq(F_46(x), F_15(x) + F_22(x) + F_47(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_45(x) + F_49(x))
Eq(F_49(x), F_21(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_15(x) + F_43(x) + F_55(x))
Eq(F_55(x), F_13(x)*F_56(x))
Eq(F_56(x), F_27(x) + F_54(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_13(x)*F_59(x))
Eq(F_59(x), F_53(x))
Eq(F_60(x), F_13(x)*F_61(x))
Eq(F_61(x), F_26(x) + F_62(x))
Eq(F_62(x), F_63(x) + F_70(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_13(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_13(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_13(x)*F_69(x))
Eq(F_69(x), F_66(x))
Eq(F_70(x), 2*F_15(x) + F_71(x) + F_79(x))
Eq(F_71(x), F_13(x)*F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_74(x) + F_76(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_13(x)*F_18(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_13(x)*F_78(x))
Eq(F_78(x), F_73(x))
Eq(F_79(x), F_13(x)*F_80(x))
Eq(F_80(x), F_62(x))
Pack JSON:
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Specification JSON:
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