0132_0321_1023_1032_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 145, 380, 982, 2511, 6366, 16026, 40106, 99859, 247541, 611255, 1504200, 3690266, 9028485, 22034006, 53652933, 130377194, 316223132, 765659945, 1850917878, 4467857616, 10770077034, 25929079371, 62350719275, 149766797709, 359368289118, 861473091150, 2063225003991, 4937161707286, 11804702048313, 28203235598296, 67332824899282, 160640525146179, 383000126904574, 912582773620070, 2173142335359514, 5172002544297251, 12302551743430497, 29248802326023331, 69503584525685604, 165082762448682754, 391922558830034825, 930059805676123030, 2206181055851650797, 5231162596230684486, 12399061007747066136, 29377796493516835901, 69581768312976687766, 164749148026023821084, 389948910248282113946, 922686410032645859355, 2182565141169094569815, 5161210056564272546841, 12201448965952745361530, 28837057429388899078806, 68135594129038876054043, 160947577304517010312694, 380089329586601802279137, 897389801880791662477620, 2118234448383378780027918, 4998819426104002801876791, 11794087373194187087989854, 27820665699414362597959346, 65611455704520559717515210, 154704657090067959202343859, 364704507697532316805376973, 859597756950127386817493775, 2025667159273053297396571456, 4772673809270104475275950186, 11242900478053145509983275533, 26480119381829048484708527446, 62357458333437306188830768917, 146820137517031285021548904354, 345631561410385317964287288404, 813529731693679481866790616145, 1914553010780395734507236753654, 4505021751191542920330290054376, 10598965088346460373777223119242, 24932658706701959729638576647531, 58642667865949749667782127735939, 137911420044935941007131649521829, 324286943016617508785604001900630, 762434799174009016248255191870430, 1792343407536093539243478025931487, 4212949070494826776102180126960086, 9901450895583489800428143562182281, 23268063522854944194301944525029776, 54672824659790355789448959125516426, 128449882825818814796466341440518059, 301750409616215797797072610221897726, 708785745482004865963249599233352830, 1664700064040619960650372485965471930, 3909415834462796006537837146437668291, 9180017221478111599683365254759392825, 21554192678239996678972851535496820731, 50603145804609277429035281878292968028
Generating function in Maple syntax:
(2*x^7-5*x^6+9*x^5-12*x^4+16*x^3-14*x^2+6*x-1)/(x-1)/(x^3-2*x^2+3*x-1)^2
Generating function in latex syntax:
\frac{2 x^{7}-5 x^{6}+9 x^{5}-12 x^{4}+16 x^{3}-14 x^{2}+6 x -1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**7 - 5*x**6 + 9*x**5 - 12*x**4 + 16*x**3 - 14*x**2 + 6*x - 1)/((x - 1)*(x**3 - 2*x**2 + 3*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-2*x^2+3*x-1)^2*F(x)-2*x^7+5*x^6-9*x^5+12*x^4-16*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)-2 x^{7}+5 x^{6}-9 x^{5}+12 x^{4}-16 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/317400*(92*(((2*n-1277/92)*23^(1/2)+I*n+63/4*I)*3^(1/2)+(6*I*n-3831/92*I)*23^(1/2)+n+63/4)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+3910*2^(2/3)*(((1/17*n-97/391)*23^(1/2)+I*n-108/17*I)*3^(1/2)+(-3/17*I*n+291/391*I)*23^(1/2)-n+108/17)*(11+3*23^(1/2)*3^(1/2))^(1/3)+59800*n)*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(-3910*2^(2/3)*(((-1/17*n+97/391)*23^(1/2)+I*n-108/17*I)*3^(1/2)+(-3/17*I*n+291/391*I)*23^(1/2)+n-108/17)*(11+3*23^(1/2)*3^(1/2))^(1/3)-92*(((-2*n+1277/92)*23^(1/2)+I*n+63/4*I)*3^(1/2)+(6*I*n-3831/92*I)*23^(1/2)-n-63/4)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+59800*n)*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1+1/317400*(-460*2^(2/3)*(3^(1/2)*(n-97/23)*23^(1/2)-17*n+108)*(11+3*23^(1/2)*3^(1/2))^(1/3)-368*(3^(1/2)*(n-1277/184)*23^(1/2)+1/2*n+63/8)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+59800*n)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(92 \left(\left(\left(2 n -\frac{1277}{92}\right) \sqrt{23}+i n +\frac{63 i}{4}\right) \sqrt{3}+\left(6 i n -\frac{3831}{92} i\right) \sqrt{23}+n +\frac{63}{4}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+3910 \,2^{\frac{2}{3}} \left(\left(\left(\frac{n}{17}-\frac{97}{391}\right) \sqrt{23}+i n -\frac{108 i}{17}\right) \sqrt{3}+\left(-\frac{3}{17} i n +\frac{291}{391} i\right) \sqrt{23}-n +\frac{108}{17}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+59800 n \right) \left(\frac{11 \left(\left(i-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 i \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{i \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(-3910 \,2^{\frac{2}{3}} \left(\left(\left(-\frac{n}{17}+\frac{97}{391}\right) \sqrt{23}+i n -\frac{108 i}{17}\right) \sqrt{3}+\left(-\frac{3}{17} i n +\frac{291}{391} i\right) \sqrt{23}+n -\frac{108}{17}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-92 \left(\left(\left(-2 n +\frac{1277}{92}\right) \sqrt{23}+i n +\frac{63 i}{4}\right) \sqrt{3}+\left(6 i n -\frac{3831}{92} i\right) \sqrt{23}-n -\frac{63}{4}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+59800 n \right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(i+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 i \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{i \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\-1+\\\frac{\left(-460 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n -\frac{97}{23}\right) \sqrt{23}-17 n +108\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-368 \left(\sqrt{3}\, \left(n -\frac{1277}{184}\right) \sqrt{23}+\frac{n}{2}+\frac{63}{8}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+59800 n \right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400} & \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) = 54
a(6) = 145
a(7) = 380
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5)-1, 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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 145
a \! \left(7\right) = 380
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19303/
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[20,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[49,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[32,x]
F[24,x] = F[25,x]+F[4,x]
F[25,x] = F[15,x]+F[26,x]+F[31,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[17,x]+F[28,x]
F[28,x] = F[18,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[11,x]*F[4,x]
F[32,x] = F[33,x]+F[48,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[26,x]
F[38,x] = F[33,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[33,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[42,x]
F[45,x] = F[43,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[45,x]
F[48,x] = F[40,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[2,x]+F[52,x]
F[52,x] = F[15,x]+F[22,x]+F[53,x]
F[53,x] = F[4,x]*F[51,x]
F[54,x] = F[52,x]+F[55,x]
F[55,x] = F[15,x]+F[44,x]+F[56,x]+F[57,x]
F[56,x] = 0
F[57,x] = F[4,x]*F[54,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_{20}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{49}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{48}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{26}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \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_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{40}\! \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_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \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_20(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_49(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_32(x))
Eq(F_24(x), F_25(x) + F_4(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_31(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_17(x) + F_28(x))
Eq(F_28(x), F_18(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_11(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_48(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_26(x))
Eq(F_38(x), F_33(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_33(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_42(x))
Eq(F_45(x), F_43(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_45(x))
Eq(F_48(x), F_40(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_2(x) + F_52(x))
Eq(F_52(x), F_15(x) + F_22(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_51(x))
Eq(F_54(x), F_52(x) + F_55(x))
Eq(F_55(x), F_15(x) + F_44(x) + F_56(x) + F_57(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_4(x)*F_54(x))
Pack JSON:
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Specification JSON:
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