0132_0213_1320_2301_3201
Counting sequence:
1, 1, 2, 6, 19, 50, 112, 232, 465, 926, 1858, 3776, 7771, 16150, 33780, 70904, 149029, 313214, 657670, 1379008, 2886831, 6033110, 12587320, 26219480, 54531769, 113252830, 234888522, 486547712, 1006642435, 2080385366, 4294979068, 8858383096, 18253625421, 37580979710, 77309428750, 158913809024, 326417535319, 670014920854, 1374389559360, 2817498572888, 5772436074721, 11819750029790, 24189255844690, 49478023286080, 101155069794219, 206708186063510, 422212465110532, 862017116223992, 1759218604492405, 3588805953115006, 7318349394534678, 14918173765726016, 30399297484815871, 61924494876413270, 126100789566446920, 256705178760195544, 522417556775059209, 1062849512059523294, 2161727821137929050, 4395513236313699968, 8935141660703165011, 18158513697557946070, 36893488147419214860, 74939897799445170680, 152185638608103923869, 308982963234635118590, 627189298506124890142, 1272825341085959203072, 2582544170319337374375, 5238875316933512813846, 10625324586456701892688, 21545797078092756456536, 43681889966544218403121, 88544371553805847940830, 179449926349046518312290, 363622219180962681654464, 736689171327664653544699, 1492267808586807887744534, 3022314549036572936990740, 6120186961799060197184248, 12391489651049949040981701, 25085210757003555375405950, 50774884423814425337921830, 102758694667243479850297280, 207935240973716218049744719, 420706185225890952798042070, 851083777008698938993451352, 1721510367131231944781908952, 3481706360490132023154112345, 7040783973435600313489105886, 14236310451781873161340277098, 28782105913385091391404998656, 58183181846412872920259211171, 117604303732111126115417186390, 237684487542793012780632248860, 480320735242727546660860609784, 970544990799738135520913815789, 1960897022228042355440213208574, 3961408125713216879677197968430, 8002044413940698096947939449728, 16162545152909924869082966348791
Generating function in Maple syntax:
(4*x^9-12*x^8+10*x^7+8*x^6-13*x^5+20*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{4 x^{9}-12 x^{8}+10 x^{7}+8 x^{6}-13 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(4*x**9 - 12*x**8 + 10*x**7 + 8*x**6 - 13*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4*F(x)+4*x^9-12*x^8+10*x^7+8*x^6-13*x^5+20*x^4-28*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+4 x^{9}-12 x^{8}+10 x^{7}+8 x^{6}-13 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,1/8*(n+2)*2^n+1/2*n^3-2*n^2+5/2*n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(n +2\right) 2^{n}}{8}+\frac{n^{3}}{2}-2 n^{2}+\frac{5 n}{2}-3 & \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) = 50
a(6) = 112
a(7) = 232
a(8) = 465
a(9) = 926
a(n+2) = -4*a(n)+4*a(n+1)+1/2*(-1+n)*(n^2-9*n+12), n >= 10
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) = 50
a \! \left(6\right) = 112
a \! \left(7\right) = 232
a \! \left(8\right) = 465
a \! \left(9\right) = 926
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(-1+n \right) \left(n^{2}-9 n +12\right)}{2}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18555/
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[24,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[23,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[17,x]*F[4,x]
F[23,x] = F[22,x]
F[24,x] = F[25,x]+F[41,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[14,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[14,x]+F[30,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[18,x]+F[37,x]
F[37,x] = F[31,x]+F[38,x]+F[40,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[11,x]+F[37,x]
F[40,x] = F[36,x]*F[4,x]
F[41,x] = F[31,x]+F[42,x]+F[63,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[59,x]
F[44,x] = F[45,x]+F[7,x]
F[45,x] = F[31,x]+F[46,x]+F[54,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[11,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[48,x]
F[51,x] = F[21,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[51,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[17,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[11,x]
F[59,x] = F[60,x]+F[72,x]
F[60,x] = F[31,x]+F[61,x]+F[63,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[60,x]+F[7,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[36,x]+F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[14,x]+F[66,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[23,x]+F[69,x]
F[72,x] = F[31,x]+F[73,x]+F[75,x]+F[82,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[45,x]+F[72,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[37,x]+F[78,x]
F[78,x] = 2*F[31,x]+F[79,x]+F[81,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[49,x]+F[78,x]
F[81,x] = F[4,x]*F[77,x]
F[82,x] = 0
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_{24}\! \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_{23}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \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_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \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_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{31}\! \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_{59}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{7}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \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_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{17}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{72}\! \left(x \right)
F_{60}\! \left(x \right) = F_{31}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \left(x \right)+F_{7}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{36}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{14}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{23}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{31}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)+F_{82}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{45}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{37}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{79}\! \left(x \right)+F_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{49}\! \left(x \right)+F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{82}\! \left(x \right) = 0
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_24(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_23(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(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_20(x)*F_4(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_17(x)*F_4(x))
Eq(F_23(x), F_22(x))
Eq(F_24(x), F_25(x) + F_41(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_14(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_14(x) + F_30(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_18(x) + F_37(x))
Eq(F_37(x), F_31(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_11(x) + F_37(x))
Eq(F_40(x), F_36(x)*F_4(x))
Eq(F_41(x), F_31(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_59(x))
Eq(F_44(x), F_45(x) + F_7(x))
Eq(F_45(x), F_31(x) + F_46(x) + F_54(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_11(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_48(x))
Eq(F_51(x), F_21(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_51(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_17(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_11(x))
Eq(F_59(x), F_60(x) + F_72(x))
Eq(F_60(x), F_31(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_60(x) + F_7(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_36(x) + F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_14(x) + F_66(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_23(x) + F_69(x))
Eq(F_72(x), F_31(x) + F_73(x) + F_75(x) + F_82(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_45(x) + F_72(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_37(x) + F_78(x))
Eq(F_78(x), 2*F_31(x) + F_79(x) + F_81(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_49(x) + F_78(x))
Eq(F_81(x), F_4(x)*F_77(x))
Eq(F_82(x), 0)
Pack JSON:
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Specification JSON:
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