0132_0213_1230_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 137, 343, 855, 2148, 5456, 13995, 36162, 93905, 244628, 638526, 1668669, 4363893, 11417347, 29879175, 78205905, 204715756, 535904882, 1402940331, 3672821924, 9615373713, 25173054502, 65903394758, 172536491655, 451705048933, 1182576987901, 3096023218695, 8105488307627, 21220434650660, 55555804233748, 145446959590331, 380785044670694, 996908126099713, 2609939255445112, 6832909513736030, 17888789081086385, 46833457398353861, 122611582578136887, 321001289469057431, 840392284426206629, 2200175561539743948, 5760134396520388342, 15080227622078976923, 39480548460101473480, 103361417742667943393, 270603704742729801546, 708449696444791420870, 1854745384525741881467, 4855786457025801620565, 12712613986379127815793, 33282055501832414078711, 87133552518666411528319, 228118602053435949887908, 597222253640458864648152, 1563548158866027199975435, 4093422222954526717735690, 10716718509992543491635441, 28056733307014998278060892, 73453481411039336401871806, 192303710926081790507801509, 503457651367171699761138133, 1318069243175377752995500907, 3450750078158871668084895463, 9034180991301091804338643705, 23651792895744168406870066028, 61921197695931032631290086266, 162111800192048313363957800395, 424414202880212910552559501932, 1111130808448588805262654549745, 2908978222465550895296314230638, 7615803858948059657656132124166, 19938433354378621244762836263919, 52199496204187793020752974830853, 136660055258184739928707440575717, 357780669570366397820701297471015, 936681953452914406699939754826195, 2452265190788376746500993220640996, 6420113618912215710191458463791772, 16808075665948270185683675981138331, 44004113378932594525858281846800814, 115204264470849512872500175736927393, 301608680033615943251249963908907072, 789621775629998315521466440712470878, 2067256646856379001112973801496199705, 5412148164939138684257496131766594213, 14169187847961037045899380205061841311, 37095415378943972444120551262667755895, 97117058288870880271382045973448616397, 254255759487668668345625265827434218700, 665650220174135124726013203069117468158, 1742694901034736705768533474109937856571, 4562434482930074992476225801550979321744, 11944608547755488271492901643563303124673, 31271391160336389821731875424449516417682, 81869564933253681193264878638116135643270, 214337303639424653757354310793540366530275
Generating function in Maple syntax:
(x^8-x^7-x^6-x^5-9*x^4+23*x^3-19*x^2+7*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{8}-x^{7}-x^{6}-x^{5}-9 x^{4}+23 x^{3}-19 x^{2}+7 x -1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**8 - x**7 - x**6 - x**5 - 9*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**4*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^2+x-1)*(x-1)^4*F(x)-x^8+x^7+x^6+x^5+9*x^4-23*x^3+19*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{8}+x^{7}+x^{6}+x^{5}+9 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(18*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-18*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)+1/30*(-12*5^(1/2)+30)*(-1/2*5^(1/2)-1/2)^(-n)+1/30*(12*5^(1/2)+30)*(1/2*5^(1/2)-1/2)^(-n)+1/6*n^3-n^2-7/6*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(18 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-18 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\\\frac{\left(-12 \sqrt{5}+30\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{30}+\frac{\left(12 \sqrt{5}+30\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-n^{2}-\\\frac{7 n}{6} & \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) = 53
a(6) = 137
a(7) = 343
a(8) = 855
a(n+4) = 1/6*n^3-n^2+a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)-25/6*n+3, 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) = 53
a \! \left(6\right) = 137
a \! \left(7\right) = 343
a \! \left(8\right) = 855
a \! \left(n +4\right) = \frac{n^{3}}{6}-n^{2}+a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-\frac{25 n}{6}+3, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18456/
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[23,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[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[43,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[37,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[28,x]
F[36,x] = F[31,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[24,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[35,x]+F[42,x]
F[42,x] = F[39,x]
F[43,x] = F[32,x]+F[44,x]+F[71,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[50,x]
F[46,x] = F[47,x]+F[6,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[10,x]
F[50,x] = F[43,x]+F[51,x]
F[51,x] = 2*F[32,x]+F[52,x]+F[56,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[47,x]
F[55,x] = F[51,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[32,x]+F[59,x]+F[69,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[10,x]+F[62,x]
F[62,x] = F[48,x]
F[63,x] = F[58,x]+F[64,x]
F[64,x] = 2*F[32,x]+F[56,x]+F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[62,x]
F[68,x] = F[64,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[24,x]+F[58,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[70,x]+F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[40,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[73,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_{23}\! \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_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{43}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{24}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)+F_{71}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{6}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)
F_{50}\! \left(x \right) = F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \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_{47}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{32}\! \left(x \right)+F_{59}\! \left(x \right)+F_{69}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{48}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{62}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{24}\! \left(x \right)+F_{58}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{40}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \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_23(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_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_43(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_37(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_24(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), F_39(x))
Eq(F_43(x), F_32(x) + F_44(x) + F_71(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_50(x))
Eq(F_46(x), F_47(x) + F_6(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_10(x))
Eq(F_50(x), F_43(x) + F_51(x))
Eq(F_51(x), 2*F_32(x) + F_52(x) + F_56(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_47(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_32(x) + F_59(x) + F_69(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_10(x) + F_62(x))
Eq(F_62(x), F_48(x))
Eq(F_63(x), F_58(x) + F_64(x))
Eq(F_64(x), 2*F_32(x) + F_56(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_62(x))
Eq(F_68(x), F_64(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_24(x) + F_58(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_70(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_40(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_73(x))
Pack JSON:
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Specification JSON:
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