0132_0231_0321_1230_2103
Counting sequence:
1, 1, 2, 6, 19, 52, 127, 298, 687, 1565, 3539, 7977, 17954, 40376, 90759, 203971, 458361, 1029974, 2314378, 5200410, 11685270, 26256622, 58998154, 132567711, 297877010, 669323637, 1503956633, 3379359954, 7593352970, 17062109331, 38338211748, 86145179928, 193566462349, 434939892958, 977301068417, 2195975567524, 4934312310593, 11087299120383, 24912934983925, 55978856777731, 125783349419100, 282632620632106, 635069733905681, 1426988739024469, 3206414591322619, 7204748187764136, 16188922227826532, 36376178052094692, 81736530144251896, 183660316112772072, 412680984317701468, 927285754603923233, 2083592177412775988, 4681789125111773871, 10519884673032400627, 23637966293763799268, 53114028135448225428, 119346137891627849637, 268168337624940125574, 602568785006059875498, 1353959769745432027079, 3042319986871983804232, 6836030958483339760195, 15360422134093231297694, 34514555239797818551507, 77553501655205528640673, 174261136416115644535319, 391561219247638999159965, 879830073256188114214742, 1976960229343899583054300, 4442189312696348281163547, 9981508781480408031166823, 22428246646313264760443069, 50395812761410589270889594, 113238363387654035271055614, 254444292890405395052557934, 571731136407054236105282074, 1284668202316859831992066658, 2886623248150368505036857646, 6486183562210542605960500067, 14574322170254593884965791318, 32748204654569361870855225257, 73584547917182775020715741965, 165342978318680318027320918070, 371522299899974049204502353054, 834803030201445641415609882423, 1875785381984185014008401200040, 4214851494269841620227909929660, 9470685340322422613048611177153, 21280436792930501832316731084146, 47816707431913584657454163416005, 107443166316435248534176446739224, 241422603271853579893490325810533, 542471665428228823663702934944515, 1218922767811875978686719748960569, 2738894597780127201143652107055351, 6154240297943901557310321028126992, 13828452425856054337077574414349006, 31072250551875883030321817749769893, 69818713231663918840410888885762041, 156881224589347666374066021106945215
Generating function in Maple syntax:
(x^7-3*x^6+3*x^5+3*x^4+3*x^2-3*x+1)/(x^2+1)/(x^3-x^2-2*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{7}-3 x^{6}+3 x^{5}+3 x^{4}+3 x^{2}-3 x +1}{\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**7 - 3*x**6 + 3*x**5 + 3*x**4 + 3*x**2 - 3*x + 1)/((x - 1)**2*(x**2 + 1)*(x**3 - x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2+1)*(x^3-x^2-2*x+1)*(x-1)^2*F(x)-x^7+3*x^6-3*x^5-3*x^4-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{7}+3 x^{6}-3 x^{5}-3 x^{4}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/2548*((-54*I*3^(1/2)-32)*(-28+84*I*3^(1/2))^(2/3)+42*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)+518*(-28+84*I*3^(1/2))^(1/3)-1372)*(1/168*(I*3^(1/2)+5)*(-28+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/2548*((11*I*3^(1/2)+97)*(-28+84*I*3^(1/2))^(2/3)-280*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-196*(-28+84*I*3^(1/2))^(1/3)-1372)*(1/84*(I*3^(1/2)-2)*(-28+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/2548*((43*I*3^(1/2)-65)*(-28+84*I*3^(1/2))^(2/3)+238*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-322*(-28+84*I*3^(1/2))^(1/3)-1372)*(1/6*(-28+84*I*3^(1/2))^(1/3)+1/3-1/168*(-28+84*I*3^(1/2))^(2/3)-1/56*I*3^(1/2)*(-28+84*I*3^(1/2))^(2/3))^(-n)-5/2*n-5/13*cos(1/2*n*Pi)+11/26*sin(1/2*n*Pi)+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-54 \,\mathrm{I} \sqrt{3}-32\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+42 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+518 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-1372\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{2548}\\+\\\frac{\left(\left(11 \,\mathrm{I} \sqrt{3}+97\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-280 \,\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-196 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-1372\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{2548}\\+\\\frac{\left(\left(43 \,\mathrm{I} \sqrt{3}-65\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+238 \,\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-322 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-1372\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}\right)^{-n}}{2548}\\-\frac{5 n}{2}-\frac{5 \cos \left(\frac{n \pi}{2}\right)}{13}+\frac{11 \sin \left(\frac{n \pi}{2}\right)}{26}+2 & \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) = 52
a(6) = 127
a(7) = 298
a(n+2) = a(n)-a(n+1)-2*a(n+4)+a(n+5)-5*n-11, n >= 8
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) = 52
a \! \left(6\right) = 127
a \! \left(7\right) = 298
a \! \left(n +2\right) = a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right)-5 n -11, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18677/
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[14,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[60,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[32,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[16,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[14,x]+F[29,x]
F[32,x] = F[33,x]+F[40,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[2,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[36,x]*F[4,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[19,x]+F[45,x]+F[53,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[11,x]+F[48,x]
F[48,x] = F[19,x]+F[24,x]+F[49,x]
F[49,x] = F[13,x]*F[4,x]
F[50,x] = F[38,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[43,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[2,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[11,x]+F[59,x]
F[59,x] = F[24,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[54,x]+F[62,x]
F[62,x] = F[44,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[14,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[31,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_{14}\! \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_{60}\! \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_{32}\! \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_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \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_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \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_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \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_{11}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{24}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{54}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{44}\! \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_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{14}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{31}\! \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) + F_14(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_60(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_32(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_16(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_14(x) + F_29(x))
Eq(F_32(x), F_33(x) + F_40(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_2(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_36(x)*F_4(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_19(x) + F_45(x) + F_53(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_11(x) + F_48(x))
Eq(F_48(x), F_19(x) + F_24(x) + F_49(x))
Eq(F_49(x), F_13(x)*F_4(x))
Eq(F_50(x), F_38(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_43(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_2(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_11(x) + F_59(x))
Eq(F_59(x), F_24(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_54(x) + F_62(x))
Eq(F_62(x), F_44(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_66(x))
Eq(F_66(x), F_14(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_31(x))
Pack JSON:
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Specification JSON:
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