0123_0231_1230_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 49, 111, 262, 646, 1640, 4231, 11003, 28721, 75096, 196496, 514314, 1346361, 3524677, 9227571, 24157930, 63246106, 165580268, 433494571, 1134903311, 2971215221, 7778742204, 20365011236, 53316291342, 139583862621, 365435296345, 956722026231, 2504730782158, 6557470320046, 17167680177776, 44945570213071, 117669030461219, 308061521170361, 806515533049632, 2111485077978296, 5527939700885010, 14472334024676481, 37889062373144173, 99194853094755771, 259695496911122866, 679891637638612546, 1779979416004714484, 4660046610375530611, 12200160415121877047, 31940434634990100221, 83621143489848423300, 218922995834555169356, 573147844013817084438, 1500520536206896083621, 3928413764606871166081, 10284720757613717414271, 26925748508234281076374, 70492524767089125814486, 184551825793033096366712, 483162952612010163285271, 1264937032042997393488715, 3311648143516982017180481, 8670007398507948658052328, 22698374052006863956976096, 59425114757512643212875546, 155576970220531065681650121, 407305795904080553832074389, 1066340417491710595814572611, 2791715456571051233611643002, 7308805952221443105020355946, 19134702400093278081449424380, 50095301248058391139327916731, 131151201344081895336534325343, 343358302784187294870275058821, 898923707008479989274290850636, 2353412818241252672952597492596, 6161314747715278029583501626654, 16130531424904581415797907386861, 42230279526998466217810220533417, 110560307156090817237632754212871, 289450641941273985495088042104670, 757791618667731139247631372100606, 1983924214061919432247806074196608, 5193981023518027157495786850488671, 13598018856492162040239554477268851, 35600075545958458963222876581317321, 93202207781383214849429075266682544, 244006547798191185585064349218729736, 638817435613190341905763972389506082, 1672445759041379840132227567949787921, 4378519841510949178490918731459857085, 11463113765491467695340528626429782731, 30010821454963453907530667147829490498, 78569350599398894027251472817058688146, 205697230343233228174223751303346573316, 538522340430300790495419781092981031171, 1409869790947669143312035591975596519559, 3691087032412706639440686994833808526861, 9663391306290450775010025392525829060372, 25299086886458645685589389182743678653596, 66233869353085486281758142155705206899750, 173402521172797813159685037284371942044981
Generating function in Maple syntax:
-(2*x^8-10*x^7+10*x^6+7*x^5-x^4+x^3-5*x^2+4*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{8}-10 x^{7}+10 x^{6}+7 x^{5}-x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**8 + 10*x**7 - 10*x**6 - 7*x**5 + x**4 - x**3 + 5*x**2 - 4*x + 1)/((x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*F(x)+2*x^8-10*x^7+10*x^6+7*x^5-x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{8}-10 x^{7}+10 x^{6}+7 x^{5}-x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,20,n = 1,13,n = 2,6,n = 4,-2,0)+1/10*(5-5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(5+5^(1/2))*(3/2+1/2*5^(1/2))^(-n)+7*n-20
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}20 & n =0 \\ 13 & n =1 \\ 6 & n =2 \\ -2 & n =4 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+7 n -20
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 49
a(6) = 111
a(7) = 262
a(8) = 646
a(n+2) = -a(n)+3*a(n+1)-7*n+13, n >= 9
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) = 49
a \! \left(6\right) = 111
a \! \left(7\right) = 262
a \! \left(8\right) = 646
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-7 n +13, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17400/
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[18,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[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]+F[39,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[38,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[11,x]+F[28,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[19,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[23,x]+F[35,x]
F[38,x] = F[22,x]*F[4,x]
F[39,x] = F[24,x]+F[40,x]+F[61,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[50,x]
F[42,x] = F[43,x]+F[7,x]
F[43,x] = F[24,x]+F[44,x]+F[45,x]
F[44,x] = F[17,x]*F[4,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[11,x]+F[48,x]
F[48,x] = F[44,x]
F[49,x] = F[43,x]
F[50,x] = F[51,x]+F[56,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[19,x]
F[55,x] = F[51,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[23,x]
F[60,x] = F[56,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[68,x]
F[63,x] = F[19,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[11,x]+F[48,x]
F[68,x] = F[69,x]+F[77,x]
F[69,x] = F[24,x]+F[25,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[76,x]
F[72,x] = F[19,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[17,x]
F[76,x] = F[69,x]
F[77,x] = F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[48,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_{18}\! \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_{7}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{24}\! \left(x \right)+F_{40}\! \left(x \right)+F_{61}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \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_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{19}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \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_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \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_{68}\! \left(x \right)
F_{63}\! \left(x \right) = F_{19}\! \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_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{11}\! \left(x \right)+F_{48}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{77}\! \left(x \right)
F_{69}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \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_{72}\! \left(x \right)+F_{76}\! \left(x \right)
F_{72}\! \left(x \right) = F_{19}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{17}\! \left(x \right)
F_{76}\! \left(x \right) = F_{69}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{48}\! \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_18(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_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_39(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_38(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_11(x) + F_28(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_19(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_23(x) + F_35(x))
Eq(F_38(x), F_22(x)*F_4(x))
Eq(F_39(x), F_24(x) + F_40(x) + F_61(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_50(x))
Eq(F_42(x), F_43(x) + F_7(x))
Eq(F_43(x), F_24(x) + F_44(x) + F_45(x))
Eq(F_44(x), F_17(x)*F_4(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_11(x) + F_48(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_43(x))
Eq(F_50(x), F_51(x) + F_56(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_19(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_23(x))
Eq(F_60(x), F_56(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_68(x))
Eq(F_63(x), F_19(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_67(x))
Eq(F_67(x), F_11(x) + F_48(x))
Eq(F_68(x), F_69(x) + F_77(x))
Eq(F_69(x), F_24(x) + F_25(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_76(x))
Eq(F_72(x), F_19(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_17(x))
Eq(F_76(x), F_69(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_48(x))
Pack JSON:
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Specification JSON:
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