0213_0231_0321_1032_1230
Counting sequence:
1, 1, 2, 6, 19, 57, 165, 467, 1303, 3600, 9874, 26927, 73086, 197585, 532346, 1430050, 3831663, 10243241, 27328509, 72781471, 193524707, 513849424, 1362645898, 3609389691, 9550748330, 25248607009, 66691801250, 176026005534, 464281361467, 1223807579865, 3224015526213, 8488976795915, 22341264603871, 58772254297488, 154548722594482, 406257799379495, 1067565671636934, 2804505093610673, 7365422361446714, 19338773373312058, 50764393275906231, 133228408871822729, 349582333691810877, 917115595325523799, 2405619958880837195, 6309052297631054992, 16544008473236538826, 43377224216841050739, 113718637906035802898, 298094888053527226177, 781329397087816801538, 2047736243428725898422, 5366328584469237259555, 14061990952739541331833, 36845576410541374004901, 96537343995229110672323, 252917599144627005259879, 662580346293958529592336, 1735704369017342879734930, 4546659512173565326533023, 11909408599614437829923022, 31193961902307412840274321, 81702402770694214775035322, 213985493352185724330140626, 560428496834070357990139839, 1467721083188850642035094377, 3843751344358339552679527485, 10065987219176709277527071887, 26360135443378728787513239923, 69028723640663899361759335120, 180760569668228754358869691594, 473336358272958356113806279147, 1239449785158134083841567150906, 3245498930125355427527887723105, 8498220151073243032835099122466, 22251993747728906871280408401870, 58264598687237854815705126702283, 152558309728867945245535971700761, 399450182924256933495003789010629, 1045888451308568048057378396548091, 2738447447955668548887038403678127, 7170014658731545467841453821792144, 18772950337539404931322664079354418, 49152104738660651348734937458889111, 128691254457290950236784287400204950, 336940708175682983628030601990127729, 882176859733547970302035012169874170, 2309700899895011650853942098967288554, 6047193887355378399066254107227274887, 15832527885848957103533613532152896585, 41451952064951051442718635930603419133, 108527100022201147844179186451839729735, 284138453722805851860116757250652512091, 743910244301720277896323645143773659408, 1947645351214516181919457131693716913802, 5099153936561654538203406216807129830115, 13350125784942261173464081405980226520354, 34951970198428582734076836242508410366593, 91507587697142208273315743691547279794434, 239575145446758938326857025813512840082278, 627228356637455120433777912081752337798451
Generating function in Maple syntax:
-(x^6-2*x^4-7*x^3+12*x^2-6*x+1)/(x^2-3*x+1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{6}-2 x^{4}-7 x^{3}+12 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**6 + 2*x**4 + 7*x**3 - 12*x**2 + 6*x - 1)/((x - 1)**2*(x**2 - 3*x + 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^2+2*x-1)*(x-1)^2*F(x)+x^6-2*x^4-7*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-2 x^{4}-7 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,(3/2-1/2*5^(1/2))^(-n)+(3/2+1/2*5^(1/2))^(-n)+1/8*(3*2^(1/2)-2)*(-1-2^(1/2))^(-n)+1/8*(-3*2^(1/2)-2)*(2^(1/2)-1)^(-n)+1/2-1/2*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\frac{\left(3 \sqrt{2}-2\right) \left(-1-\sqrt{2}\right)^{-n}}{8}+\\\frac{\left(-3 \sqrt{2}-2\right) \left(\sqrt{2}-1\right)^{-n}}{8}+\frac{1}{2}-\frac{n}{2} & \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) = 57
a(6) = 165
a(n+4) = a(n)-a(n+1)-6*a(n+2)+5*a(n+3)-n, n >= 7
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) = 57
a \! \left(6\right) = 165
a \! \left(n +4\right) = a \! \left(n \right)-a \! \left(n +1\right)-6 a \! \left(n +2\right)+5 a \! \left(n +3\right)-n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20236/
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[10,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = F[9,x]
F[14,x] = F[15,x]+F[37,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[1,x]+F[10,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[35,x]
F[24,x] = 0
F[25,x] = F[10,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = x^2
F[30,x] = F[31,x]+F[33,x]
F[31,x] = F[24,x]+F[25,x]+F[32,x]
F[32,x] = F[10,x]*F[15,x]
F[33,x] = F[34,x]
F[34,x] = F[10,x]*F[31,x]
F[35,x] = F[10,x]*F[36,x]
F[36,x] = F[15,x]+F[31,x]
F[37,x] = F[24,x]+F[38,x]+F[72,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[40,x]+F[50,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[1,x]+F[41,x]
F[44,x] = F[24,x]+F[45,x]+F[48,x]
F[45,x] = F[10,x]*F[46,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[10,x]*F[49,x]
F[49,x] = F[10,x]+F[47,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[24,x]+F[38,x]+F[52,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[15,x]+F[51,x]
F[54,x] = F[24,x]+F[55,x]+F[68,x]+F[70,x]
F[55,x] = F[10,x]*F[56,x]
F[56,x] = F[57,x]+F[62,x]
F[57,x] = F[58,x]+F[60,x]
F[58,x] = F[59,x]
F[59,x] = F[10,x]*F[41,x]
F[60,x] = F[61,x]
F[61,x] = F[10,x]*F[47,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = 2*F[24,x]+F[55,x]+F[64,x]
F[64,x] = F[10,x]*F[51,x]
F[65,x] = F[66,x]
F[66,x] = F[10,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[10,x]*F[69,x]
F[69,x] = F[31,x]+F[67,x]
F[70,x] = F[10,x]*F[71,x]
F[71,x] = F[51,x]+F[67,x]
F[72,x] = F[10,x]*F[73,x]
F[73,x] = F[14,x]+F[74,x]
F[74,x] = F[31,x]+F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[10,x]*F[77,x]
F[77,x] = F[74,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{10}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \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_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = x^{2}
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{72}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{24}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{24}\! \left(x \right)+F_{55}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{55}\! \left(x \right) = F_{10}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{62}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right) F_{41}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{55}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right) F_{51}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{10}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{10}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{51}\! \left(x \right)+F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = F_{10}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{14}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{31}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{10}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{74}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_14(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x))
Eq(F_13(x), F_9(x))
Eq(F_14(x), F_15(x) + F_37(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_10(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_35(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_10(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), x**2)
Eq(F_30(x), F_31(x) + F_33(x))
Eq(F_31(x), F_24(x) + F_25(x) + F_32(x))
Eq(F_32(x), F_10(x)*F_15(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_10(x)*F_31(x))
Eq(F_35(x), F_10(x)*F_36(x))
Eq(F_36(x), F_15(x) + F_31(x))
Eq(F_37(x), F_24(x) + F_38(x) + F_72(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_50(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_1(x) + F_41(x))
Eq(F_44(x), F_24(x) + F_45(x) + F_48(x))
Eq(F_45(x), F_10(x)*F_46(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_10(x)*F_49(x))
Eq(F_49(x), F_10(x) + F_47(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_24(x) + F_38(x) + F_52(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_15(x) + F_51(x))
Eq(F_54(x), F_24(x) + F_55(x) + F_68(x) + F_70(x))
Eq(F_55(x), F_10(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_62(x))
Eq(F_57(x), F_58(x) + F_60(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_10(x)*F_41(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_10(x)*F_47(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), 2*F_24(x) + F_55(x) + F_64(x))
Eq(F_64(x), F_10(x)*F_51(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_10(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_10(x)*F_69(x))
Eq(F_69(x), F_31(x) + F_67(x))
Eq(F_70(x), F_10(x)*F_71(x))
Eq(F_71(x), F_51(x) + F_67(x))
Eq(F_72(x), F_10(x)*F_73(x))
Eq(F_73(x), F_14(x) + F_74(x))
Eq(F_74(x), F_31(x) + F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_10(x)*F_77(x))
Eq(F_77(x), F_74(x))
Pack JSON:
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Specification JSON:
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