0123_1032_1302_1320
Counting sequence:
1, 1, 2, 6, 20, 62, 183, 522, 1454, 3983, 10782, 28938, 77183, 204914, 542166, 1430799, 3768662, 9912050, 26041215, 68359050, 179330846, 470223311, 1232518734, 3229692186, 8461276415, 22163574242, 58048320678, 152019136527, 398084586374, 1042385617538, 2729374256127, 7146341130618, 18710857095278, 48988646074319, 128259912965886, 335800756499754, 879161683886207, 2301722949864530, 6026084475118710, 15776685094314255, 41304280045469366, 108136773517384466, 283107277457265279, 741187532755573866, 1940460268611781310, 5080203168684420047, 13300169028650778798, 34820343499686516282, 91160940635245969919, 238662636735725793218, 624827286231280209222, 1635819855276812433423, 4282633546236552288998, 11212083316707634829474, 29353621470435932991231, 76848791227699325727834, 201192772478860367359502, 526729566741278422685135, 1378996008809768193364830, 3610258621817612742747210, 9451780180902243205552511, 24745082569407463215261746, 64783468824356839122935574, 169605326497736439518950671, 444032515856999250164727830, 1162492231449554852436855602, 3043444199244252390069084543, 7967840407788376483616889162, 20860077107131225392474565214, 54612391079625996357192771023, 142977096463788157005875676942, 374318898975821261313978117978, 979979601791841200243146393343, 2565619909056033486029636494754, 6716880130688921551074113956326, 17585020493636055753649407105039, 46038181371469894882787510820422, 120529523663274927240539932279490, 315550389703357483530485899864575, 826121645616802716734224995007290, 2162814547487061053438803540543406, 5662321997524401217115414537395151, 14824151446446184144973897893186494, 38810132344534234311939194785253226, 101606245592596684979109517748750975, 266008604444136153001921021033355282, 696419567761572438779716870496026038, 1823250098884102492843356240744145167, 4773330728977777698762605152315254134, 12496742088223315921468965817359306578, 32716895536040340701693305502078044287, 85653944520594047455708977093505583658, 224244938027134484209629678587700221438, 587080869563594770261572164288118110159, 1536997670669220556751871025513700168046, 4023912142455208360347609334727074511994, 10534738756718687444998093823615707603967, 27580304127745419816060945826016416771970, 72206173626606703686013291034226279656070, 189038216752252954607636022036247896084495, 494908476630508686868208964593688356373926
Generating function in Maple syntax:
(x^6-2*x^5+x^4-3*x^3+8*x^2-5*x+1)/(x-1)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{6}-2 x^{5}+x^{4}-3 x^{3}+8 x^{2}-5 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**6 - 2*x**5 + x**4 - 3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^2-3*x+1)*F(x)+x^6-2*x^5+x^4-3*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{6}-2 x^{5}+x^{4}-3 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/40*(32*5^(1/2)-40)*(3/2-1/2*5^(1/2))^(-n)+1/40*(-32*5^(1/2)-40)*(3/2+1/2*5^(1/2))^(-n)-9/8*2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(32 \sqrt{5}-40\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(-32 \sqrt{5}-40\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}-\frac{9 \,2^{n}}{8}+1 & \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) = 183
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1, 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 183
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15155/
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[26,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[24,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[11,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[7,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[15,x]+F[28,x]+F[73,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[1,x]+F[4,x]
F[34,x] = F[15,x]+F[16,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[22,x]+F[7,x]
F[37,x] = F[38,x]+F[64,x]
F[38,x] = F[15,x]+F[28,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[15,x]+F[42,x]+F[63,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[22,x]+F[4,x]
F[45,x] = F[41,x]+F[46,x]
F[46,x] = F[15,x]+F[47,x]+F[60,x]+F[62,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[56,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[4,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[22,x]
F[56,x] = F[57,x]
F[57,x] = 2*F[15,x]+F[47,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[41,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[41,x]
F[62,x] = 0
F[63,x] = F[2,x]*F[4,x]
F[64,x] = F[15,x]+F[60,x]+F[65,x]+F[72,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[51,x]
F[69,x] = F[54,x]
F[70,x] = F[71,x]
F[71,x] = 2*F[15,x]+F[58,x]+F[65,x]
F[72,x] = 0
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[2,x]+F[76,x]
F[76,x] = F[15,x]+F[42,x]+F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[2,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_{26}\! \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_{24}\! \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_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{73}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \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_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{64}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{63}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \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_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \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_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{58}\! \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_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{41}\! \left(x \right)
F_{62}\! \left(x \right) = 0
F_{63}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)+F_{72}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{51}\! \left(x \right)
F_{69}\! \left(x \right) = F_{54}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{72}\! \left(x \right) = 0
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{2}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{2}\! \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_26(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_24(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_11(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_7(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_15(x) + F_28(x) + F_73(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_37(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_1(x) + F_4(x))
Eq(F_34(x), F_15(x) + F_16(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_22(x) + F_7(x))
Eq(F_37(x), F_38(x) + F_64(x))
Eq(F_38(x), F_15(x) + F_28(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_63(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_22(x) + F_4(x))
Eq(F_45(x), F_41(x) + F_46(x))
Eq(F_46(x), F_15(x) + F_47(x) + F_60(x) + F_62(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_56(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_4(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_22(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), 2*F_15(x) + F_47(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_41(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_41(x))
Eq(F_62(x), 0)
Eq(F_63(x), F_2(x)*F_4(x))
Eq(F_64(x), F_15(x) + F_60(x) + F_65(x) + F_72(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_51(x))
Eq(F_69(x), F_54(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), 2*F_15(x) + F_58(x) + F_65(x))
Eq(F_72(x), 0)
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_2(x) + F_76(x))
Eq(F_76(x), F_15(x) + F_42(x) + F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_2(x))
Pack JSON:
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Specification JSON:
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