0123_0132_1032_1320_3102
Counting sequence:
1, 1, 2, 6, 19, 50, 110, 240, 513, 1066, 2169, 4342, 8581, 16784, 32552, 62692, 120030, 228666, 433774, 819850, 1544645, 2902190, 5439731, 10174484, 18995085, 35404560, 65894442, 122484690, 227416006, 421814478, 781684644, 1447418172, 2678221853, 4952478546, 9152755321, 16906770930, 31215659649, 57611383432, 106288596752, 196030777928, 361439767122, 666246534898, 1227815816214, 2262253827742, 4167434538929, 7675807746910, 14135619446419, 26028423532296, 47921372170409, 88219469609024, 162389648743394, 298892707156446, 550097011843178, 1012357135722030, 1862952760315084, 3428043897743912, 6307682956809353, 11605785351622746, 21353172957392457, 39285773189091566, 72276115459320109, 132966534212310528, 244612827437197864, 449993859022592524, 827800059901878486, 1522778524975530314, 2801181671953498062, 5152758431178442610, 9478353808398538765, 17434972211739295822, 32070470655276409507, 58990978848613653788, 108508180485279610261, 199588878993540504528, 367119544555466729962, 675268165782223686506, 1242060962276745652798, 2284586718792965620878, 4202141682430602150308, 7729158773197211050484, 14216491364090667429765, 26148779891419291541378, 48096045706806218008233, 88463958601178409020714, 162713102822155920209865, 299280166586607875675192, 550468766589805351348784, 1012480893730396533235824, 1862260643414641020194474, 3425260657866139189328002, 6300084465738082653343830, 11587740171856481062714950, 21313304851092421386637081, 39201488109202932159325118, 72103118848413304892903139, 132618868345646490901597328, 243925037296736250569006737, 448649574961642036431520256, 825197644749053145443641602, 1517779055244449279902121910, 2791637372639043021483500418
Generating function in Maple syntax:
(2*x^14+8*x^13+7*x^12-10*x^11-21*x^10-3*x^9+22*x^8+7*x^7-18*x^6-6*x^5+3*x^4+5*x^3+x^2-3*x+1)/(x-1)/(x^3+x^2+x-1)/(x^2+x-1)^2
Generating function in latex syntax:
\frac{2 x^{14}+8 x^{13}+7 x^{12}-10 x^{11}-21 x^{10}-3 x^{9}+22 x^{8}+7 x^{7}-18 x^{6}-6 x^{5}+3 x^{4}+5 x^{3}+x^{2}-3 x +1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**14 + 8*x**13 + 7*x**12 - 10*x**11 - 21*x**10 - 3*x**9 + 22*x**8 + 7*x**7 - 18*x**6 - 6*x**5 + 3*x**4 + 5*x**3 + x**2 - 3*x + 1)/((x - 1)*(x**2 + x - 1)**2*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^3+x^2+x-1)*(x^2+x-1)^2*F(x)+2*x^14+8*x^13+7*x^12-10*x^11-21*x^10-3*x^9+22*x^8+7*x^7-18*x^6-6*x^5+3*x^4+5*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{14}+8 x^{13}+7 x^{12}-10 x^{11}-21 x^{10}-3 x^{9}+22 x^{8}+7 x^{7}-18 x^{6}-6 x^{5}+3 x^{4}+5 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
-1/5*(n+1)*Sum(_alpha^(-2-n),_alpha = RootOf(_Z^2+_Z-1))-64/25*Sum(_alpha^(-n),_alpha = RootOf(_Z^2+_Z-1))-132/25*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^2+_Z-1))-28/11*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-2*_Z+1))-19/22*Sum(_alpha^(1-n),_alpha = RootOf(_Z^4-2*_Z+1))+1/11*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-2*_Z+1))+64/11*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-2*_Z+1))+piecewise(Or(n = 1,n = 4),1,n = 5,4,n = 6,2,0)
Explicit closed form in latex syntax:
-\frac{\left(n +1\right) \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-2-n}\right)}{5}-\frac{64 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{25}-\frac{132 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{25}-\frac{28 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{11}-\frac{19 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{22}+\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{11}+\frac{64 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{11}+\left(\left\{\begin{array}{cc}1 & n =1\text{ or } n =4 \\ 4 & n =5 \\ 2 & n =6 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\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) = 110
a(7) = 240
a(8) = 513
a(9) = 1066
a(10) = 2169
a(11) = 4342
a(12) = 8581
a(13) = 16784
a(14) = 32552
a(n+4) = 1/4*a(n)+3/4*a(n+1)+1/2*a(2+n)-1/2*a(n+3)+3/4*a(n+6)-1/4*a(n+7)-5/4, n >= 15
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) = 110
a \! \left(7\right) = 240
a \! \left(8\right) = 513
a \! \left(9\right) = 1066
a \! \left(10\right) = 2169
a \! \left(11\right) = 4342
a \! \left(12\right) = 8581
a \! \left(13\right) = 16784
a \! \left(14\right) = 32552
a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(2+n \right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}-\frac{5}{4}, \quad n \geq 15
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16377/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[18,x]
F[24,x] = F[25,x]+F[36,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[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[19,x]+F[37,x]+F[57,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[41,x]
F[41,x] = F[10,x]*F[4,x]
F[42,x] = F[19,x]+F[20,x]+F[43,x]
F[43,x] = F[11,x]*F[4,x]
F[44,x] = F[45,x]+F[56,x]
F[45,x] = F[19,x]+F[37,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[25,x]+F[48,x]
F[48,x] = F[19,x]+F[35,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[18,x]+F[4,x]
F[52,x] = F[48,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[48,x]
F[56,x] = F[54,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[47,x]+F[59,x]
F[59,x] = F[60,x]+F[67,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[40,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[22,x]
F[67,x] = 2*F[19,x]+F[68,x]+F[71,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[22,x]*F[4,x]
F[71,x] = F[4,x]*F[60,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_{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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \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_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \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_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{56}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{25}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \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_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \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_{48}\! \left(x \right)
F_{56}\! \left(x \right) = F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{47}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{67}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{40}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{22}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \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_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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_18(x))
Eq(F_24(x), F_25(x) + F_36(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_12(x))
Eq(F_29(x), F_25(x) + F_30(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))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_57(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_10(x)*F_4(x))
Eq(F_42(x), F_19(x) + F_20(x) + F_43(x))
Eq(F_43(x), F_11(x)*F_4(x))
Eq(F_44(x), F_45(x) + F_56(x))
Eq(F_45(x), F_19(x) + F_37(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_25(x) + F_48(x))
Eq(F_48(x), F_19(x) + F_35(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_18(x) + F_4(x))
Eq(F_52(x), F_48(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_48(x))
Eq(F_56(x), F_54(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_47(x) + F_59(x))
Eq(F_59(x), F_60(x) + F_67(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_40(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_22(x))
Eq(F_67(x), 2*F_19(x) + F_68(x) + F_71(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_22(x)*F_4(x))
Eq(F_71(x), F_4(x)*F_60(x))
Pack JSON:
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Specification JSON:
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