0123_0321_1230_3021_3102
Counting sequence:
1, 1, 2, 6, 19, 47, 85, 139, 226, 367, 595, 964, 1561, 2527, 4090, 6619, 10711, 17332, 28045, 45379, 73426, 118807, 192235, 311044, 503281, 814327, 1317610, 2131939, 3449551, 5581492, 9031045, 14612539, 23643586, 38256127, 61899715, 100155844, 162055561, 262211407, 424266970, 686478379, 1110745351, 1797223732, 2907969085, 4705192819, 7613161906, 12318354727, 19931516635, 32249871364, 52181388001, 84431259367, 136612647370, 221043906739, 357656554111, 578700460852, 936357014965, 1515057475819, 2451414490786, 3966471966607, 6417886457395, 10384358424004, 16802244881401, 27186603305407, 43988848186810, 71175451492219, 115164299679031, 186339751171252, 301504050850285, 487843802021539, 789347852871826, 1277191654893367, 2066539507765195, 3343731162658564, 5410270670423761, 8754001833082327, 14164272503506090, 22918274336588419, 37082546840094511, 60000821176682932, 97083368016777445, 157084189193460379, 254167557210237826, 411251746403698207, 665419303613936035, 1076671050017634244, 1742090353631570281, 2818761403649204527, 4560851757280774810, 7379613160929979339, 11940464918210754151, 19320078079140733492, 31260542997351487645, 50580621076492221139, 81841164073843708786, 132421785150335929927, 214262949224179638715, 346684734374515568644, 560947683598695207361, 907632417973210776007, 1468580101571905983370, 2376212519545116759379, 3844792621117022742751
Generating function in Maple syntax:
-(x+1)*(5*x^7+7*x^6-4*x^5-7*x^4-x^3-2*x^2+2*x-1)/(x-1)/(x^2+x-1)
Generating function in latex syntax:
-\frac{\left(x +1\right) \left(5 x^{7}+7 x^{6}-4 x^{5}-7 x^{4}-x^{3}-2 x^{2}+2 x -1\right)}{\left(x -1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x - 1)*(5*x**7 + 7*x**6 - 4*x**5 - 7*x**4 - x**3 - 2*x**2 + 2*x - 1)/((x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+x-1)*F(x)+(x+1)*(5*x^7+7*x^6-4*x^5-7*x^4-x^3-2*x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+\left(x +1\right) \left(5 x^{7}+7 x^{6}-4 x^{5}-7 x^{4}-x^{3}-2 x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-piecewise(n = 1,6,n = 2,8,n = 3,13,n = 4,12,n = 5,5,0)+1/2*(-3*5^(1/2)+3)*(-1/2*5^(1/2)-1/2)^(-n)-2+1/2*(3*5^(1/2)+3)*(1/2*5^(1/2)-1/2)^(-n)
Explicit closed form in latex syntax:
-\left(\left\{\begin{array}{cc}6 & n =1 \\ 8 & n =2 \\ 13 & n =3 \\ 12 & n =4 \\ 5 & n =5 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(-3 \sqrt{5}+3\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}-2+\frac{\left(3 \sqrt{5}+3\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 47
a(6) = 85
a(7) = 139
a(8) = 226
a(n+2) = a(n)+a(1+n)+2, 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) = 47
a \! \left(6\right) = 85
a \! \left(7\right) = 139
a \! \left(8\right) = 226
a \! \left(n +2\right) = a \! \left(n \right)+a \! \left(1+n \right)+2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17806/
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[29,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[24,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[4,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[4,x]
F[29,x] = F[30,x]+F[38,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[15,x]
F[34,x] = F[30,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[30,x]
F[38,x] = F[18,x]+F[39,x]+F[69,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[54,x]
F[41,x] = F[42,x]+F[7,x]
F[42,x] = F[18,x]+F[19,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[15,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[11,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[27,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[21,x]
F[54,x] = F[38,x]+F[55,x]
F[55,x] = 2*F[18,x]+F[56,x]+F[66,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[11,x]+F[46,x]
F[62,x] = F[58,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[58,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[35,x]+F[63,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[30,x]+F[58,x]
F[72,x] = F[73,x]+F[78,x]
F[73,x] = F[18,x]+F[36,x]+F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[15,x]+F[50,x]
F[77,x] = F[35,x]
F[78,x] = 2*F[18,x]+F[56,x]+F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[81,x]+F[82,x]
F[81,x] = F[17,x]+F[51,x]
F[82,x] = F[56,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_{29}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \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_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \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_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{69}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \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_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{27}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{21}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{66}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{58}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{35}\! \left(x \right)+F_{63}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{30}\! \left(x \right)+F_{58}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{78}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{15}\! \left(x \right)+F_{50}\! \left(x \right)
F_{77}\! \left(x \right) = F_{35}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{82}\! \left(x \right)
F_{81}\! \left(x \right) = F_{17}\! \left(x \right)+F_{51}\! \left(x \right)
F_{82}\! \left(x \right) = F_{56}\! \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_29(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_24(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_4(x))
Eq(F_29(x), F_30(x) + F_38(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_34(x))
Eq(F_33(x), F_1(x) + F_15(x))
Eq(F_34(x), F_30(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_30(x))
Eq(F_38(x), F_18(x) + F_39(x) + F_69(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_54(x))
Eq(F_41(x), F_42(x) + F_7(x))
Eq(F_42(x), F_18(x) + F_19(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_15(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_11(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_27(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_21(x))
Eq(F_54(x), F_38(x) + F_55(x))
Eq(F_55(x), 2*F_18(x) + F_56(x) + F_66(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_11(x) + F_46(x))
Eq(F_62(x), F_58(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_58(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_35(x) + F_63(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_30(x) + F_58(x))
Eq(F_72(x), F_73(x) + F_78(x))
Eq(F_73(x), F_18(x) + F_36(x) + F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_15(x) + F_50(x))
Eq(F_77(x), F_35(x))
Eq(F_78(x), 2*F_18(x) + F_56(x) + F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_82(x))
Eq(F_81(x), F_17(x) + F_51(x))
Eq(F_82(x), F_56(x))
Pack JSON:
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Specification JSON:
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