0123_0231_0312_1302_2103

Counting sequence:
1, 1, 2, 6, 19, 53, 138, 364, 981, 2657, 7166, 19270, 51823, 139485, 375542, 1010964, 2721185, 7324449, 19715370, 53069046, 142848731, 384511445, 1035003154, 2785957644, 7499074013, 20185558033, 54334264806, 146253679574, 393676798551, 1059675398941, 2852370163294, 7677837476564, 20666738446601, 55629476349489, 149740059400114, 403061234189670, 1084935848957539, 2920364689835445, 7860861017627226, 21159390180396012, 56955566547866533, 153309548768146561, 412669369614300686, 1110798446581486374, 2989980065840892031, 8048247476079818909, 21663785714235960966, 58313267934052495828, 156964127230720764465, 422506542853446599681, 1137277554454109179514, 3061254926681109678742, 8240100835039746743851, 22180204980588194804949, 59703334076801642893922, 160705823188098577575948, 432578213694760968101165, 1164387868785131146740657, 3134228830882982289170550, 8436527576148127496244534, 22708934600367458399965415, 61126536484246042038919517, 164536713338344020387366766, 442889972069031996368614932, 1192144436214384570526749977, 3208942280082587089141988497, 8637636719255599305510784386, 23250268026607340804717248646, 62583665055511495648588285491, 168458923884155085792567413557, 453447541160077563554581008682, 1220562662061857957917163323244, 3285436741388359547366923315829, 8843539883016279924792461347233, 23804505707649201380636997925150, 64075528519269104652780853863942, 172474631712448252519562259689359, 464256780580418434420309047901533, 1249658318878084172499333940544790, 3363754670397736109185902882032532, 9054351346839192310849852497892801, 24371955253893438405651347306368993, 65602954882587177260505329037543626, 176586065602563495223129421986517814, 475323689623462152656417094991825531, 1279447555198157857353435187730462933, 3443939534760652842759197846603987634, 9270188114315057905002148254719047180, 24952931608527959342866289495296999293, 67166791890487876638383431483871057489, 180795507463271019119740206508016043014, 486654410593202733611245107782200215958, 1309946904504419546119104394345712746615, 3526035838304756682822315667926241971357, 9491169978155081438763099902669026040702, 25547757222326473509559585827732894248532, 68767907496458690350526266891516420976681, 185105293599265571037349390006981564743153, 498255232213293441336988214480619535373394, 1341173294402791551239115623260943543430822, 3610089145735711294801681407532498794129155

Generating function in Maple syntax:
(x-1)^2/(2*x^2-x+1)/(x^4-2*x^2-2*x+1)

Generating function in latex syntax:
\frac{\left(x -1\right)^{2}}{\left(2 x^{2}-x +1\right) \left(x^{4}-2 x^{2}-2 x +1\right)}

Generating function in sympy syntax:
(x - 1)**2/((2*x**2 - x + 1)*(x**4 - 2*x**2 - 2*x + 1))

Implicit equation for the generating function in Maple syntax:
(2*x^2-x+1)*(x^4-2*x^2-2*x+1)*F(x)-(x-1)^2 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x^{2}-x +1\right) \left(x^{4}-2 x^{2}-2 x +1\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0

Explicit closed form in Maple syntax:
-9524/255619*Sum(_alpha^(-n+4),_alpha = RootOf(2*_Z^6-_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1))-6142/255619*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^6-_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1))+1010/36517*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^6-_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1))+20149/255619*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^6-_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1))+82317/511238*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^6-_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1))+31547/511238*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^6-_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1))

Explicit closed form in latex syntax:
-\frac{9524 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{255619}-\frac{6142 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{255619}+\frac{1010 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{36517}+\frac{20149 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{255619}+\frac{82317 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{511238}+\frac{31547 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{511238}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(n+6) = -2*a(n)+a(n+1)+3*a(n+2)+2*a(n+3)-2*a(n+4)+3*a(n+5), n >= 6

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) = 53
a \! \left(n +6\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 a \! \left(n +3\right)-2 a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17037/
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[17,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[55,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[30,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[27,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[13,x]+F[32,x]
F[32,x] = F[27,x]
F[33,x] = F[34,x]+F[44,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[37,x]*F[4,x]
F[40,x] = F[38,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[38,x]
F[44,x] = 2*F[19,x]+F[45,x]+F[52,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[38,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[38,x]+F[51,x]
F[51,x] = F[49,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[40,x]+F[54,x]
F[54,x] = F[51,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[65,x]
F[57,x] = F[2,x]+F[58,x]
F[58,x] = F[19,x]+F[39,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[11,x]+F[62,x]
F[62,x] = F[19,x]+F[24,x]+F[28,x]
F[63,x] = F[38,x]+F[64,x]
F[64,x] = 2*F[19,x]+F[45,x]+F[49,x]
F[65,x] = F[38,x]+F[66,x]
F[66,x] = F[42,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_{17}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{55}\! \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_{33}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \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_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{65}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \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_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{63}\! \left(x \right) = F_{38}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{65}\! \left(x \right) = F_{38}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{42}\! \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_17(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_55(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_30(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x) + F_27(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_13(x) + F_32(x))
Eq(F_32(x), F_27(x))
Eq(F_33(x), F_34(x) + F_44(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_37(x)*F_4(x))
Eq(F_40(x), F_38(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_38(x))
Eq(F_44(x), 2*F_19(x) + F_45(x) + F_52(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_38(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_38(x) + F_51(x))
Eq(F_51(x), F_49(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_40(x) + F_54(x))
Eq(F_54(x), F_51(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_65(x))
Eq(F_57(x), F_2(x) + F_58(x))
Eq(F_58(x), F_19(x) + F_39(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_11(x) + F_62(x))
Eq(F_62(x), F_19(x) + F_24(x) + F_28(x))
Eq(F_63(x), F_38(x) + F_64(x))
Eq(F_64(x), 2*F_19(x) + F_45(x) + F_49(x))
Eq(F_65(x), F_38(x) + F_66(x))
Eq(F_66(x), F_42(x))
Pack JSON:
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Specification JSON:
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