0213_0321_1032_1230
Counting sequence:
1, 1, 2, 6, 20, 61, 177, 499, 1384, 3800, 10362, 28111, 75951, 204509, 549070, 1470434, 3929168, 10478649, 27896845, 74153567, 196837252, 521846628, 1381952870, 3656000859, 9663277659, 25520276857, 67347670298, 177609413502, 468104046476, 1233036357877, 3246295767273, 8542766056075, 22471123365280, 59085761080496, 155305594921938, 408085050817447, 1071977046840327, 2815155095455445, 7391133740339686, 19400846132942810, 50914250174060744, 133590195427762545, 350455763701845061, 919224241901532023, 2410710682042887868, 6321342390531164604, 16573679382198808766, 43448856127665700275, 113891572636647371955, 298512389425575013873, 782337334562523946034, 2050169619750187975158, 5372203274586868557572, 14076173709296266004653, 36879816613772454648609, 96620007158247996632611, 253117165673895857824216, 663062142515515120681352, 1736867527989724914477354, 4549467626339885987106943, 11916187986919461185813343, 31210328791083780212628941, 81741915935551972875634942, 214080886570677607903694546, 560658796435911883237847360, 1468277075611025576104063401, 3845093628804530946065173117, 10069227780491266998367332239, 26367958850454035622579406324, 69047611016129070752731928340, 180806167826234403975881044502, 473446441964434826738801578251, 1239715550699092674708569102091, 3246140544898749079886886924649, 8499769146160988928420099476810, 22255733352677792314809408312174, 58273626892223371598348126877308, 152580105743787864254350971961189, 399502803159082288295276789706585, 1046015487793138676666739398200507, 2738754141159635160906033407678992, 7170755081624049320488804831446368, 18774737876528379248636361102663810, 49156420239531103836009682515162199, 128701672998020829528647474536060599, 336965860758013194699031721318112197, 882237583438938271735900437961698838, 2309847499888122464792674069878922442, 6047547811046990328377583474842367416, 15833382333225291776095004238294715617, 41454014883395332717152746710502149813, 108532080106466045065608798717779010215, 284150476709779927577410092562429803820, 743939270359933326552339928033267523436, 1947715426317916354948783032784481933678, 5099323112826667932918074301878153733987, 13350534212575688135922743477213039348067, 34952956229960450053708828470045059925985, 91509968187839369875038390217853391741026, 239580892459685128849934311093661713534950, 627242231154004663081655129168356196650484
Generating function in Maple syntax:
-(2*x-1)*(x^5-x^4-x^3-4*x^2+4*x-1)/(x^2-3*x+1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{5}-x^{4}-x^{3}-4 x^{2}+4 x -1\right)}{\left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(1 - 2*x)*(x**5 - x**4 - x**3 - 4*x**2 + 4*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)+(2*x-1)*(x^5-x^4-x^3-4*x^2+4*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)+\left(2 x -1\right) \left(x^{5}-x^{4}-x^{3}-4 x^{2}+4 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1-n+1/2*(-1-2^(1/2))^(-n)*2^(1/2)-1/2*(2^(1/2)-1)^(-n)*2^(1/2)+(3/2+1/2*5^(1/2))^(-n)+(3/2-1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1-n +\frac{\left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{2}-\frac{\left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{2}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} & \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) = 61
a(6) = 177
a(n+4) = a(n)-a(n+1)-6*a(n+2)+5*a(n+3)-2*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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 177
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)-2 n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15731/
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[12,x]*F[4,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[1,x]+F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[39,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[37,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[26,x]+F[27,x]+F[34,x]
F[34,x] = F[12,x]*F[18,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[33,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[18,x]+F[33,x]
F[39,x] = F[26,x]+F[40,x]+F[74,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[52,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[1,x]+F[43,x]
F[46,x] = F[26,x]+F[47,x]+F[50,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[43,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[12,x]+F[49,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[26,x]+F[40,x]+F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[18,x]+F[53,x]
F[56,x] = F[26,x]+F[57,x]+F[70,x]+F[72,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[64,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[43,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[49,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = 2*F[26,x]+F[57,x]+F[66,x]
F[66,x] = F[12,x]*F[53,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[33,x]+F[69,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[53,x]+F[69,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[17,x]+F[76,x]
F[76,x] = F[77,x]+F[79,x]
F[77,x] = F[26,x]+F[27,x]+F[78,x]
F[78,x] = F[12,x]*F[38,x]
F[79,x] = F[80,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[76,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \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_{12}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)+F_{74}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{26}\! \left(x \right)+F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{26}\! \left(x \right)+F_{57}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{57}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{33}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{53}\! \left(x \right)+F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{17}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{76}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_17(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_12(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_39(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_37(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_26(x) + F_27(x) + F_34(x))
Eq(F_34(x), F_12(x)*F_18(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_33(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_18(x) + F_33(x))
Eq(F_39(x), F_26(x) + F_40(x) + F_74(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_52(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_1(x) + F_43(x))
Eq(F_46(x), F_26(x) + F_47(x) + F_50(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_43(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_12(x) + F_49(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_26(x) + F_40(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_18(x) + F_53(x))
Eq(F_56(x), F_26(x) + F_57(x) + F_70(x) + F_72(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_64(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_43(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_49(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), 2*F_26(x) + F_57(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_53(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_33(x) + F_69(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_53(x) + F_69(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_17(x) + F_76(x))
Eq(F_76(x), F_77(x) + F_79(x))
Eq(F_77(x), F_26(x) + F_27(x) + F_78(x))
Eq(F_78(x), F_12(x)*F_38(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_76(x))
Pack JSON:
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Specification JSON:
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