0123_0231_1032_2031_3021
Counting sequence:
1, 1, 2, 6, 19, 54, 147, 392, 1035, 2720, 7133, 18688, 48941, 128146, 335509, 878394, 2299687, 6020682, 15762375, 41266460, 108037023, 282844628, 740496881, 1938646036, 5075441249, 13287677734, 34787591977, 91075098222, 238437702715, 624238009950, 1634276327163, 4278590971568, 11201496587571, 29325898791176, 76776199785989, 201002700566824, 526231901914517, 1377693005176762, 3606847113615805, 9442848335670690, 24721697893396303, 64722245344518258, 169445038140158511, 443612869075957316, 1161393569087713479, 3040567838187183164, 7960309945473836057, 20840361998234325052, 54560776049229139145, 142841966149453092430, 373965122399130138193, 979053401047937322198, 2563195080744681828451, 6710531841186108163206, 17568400442813642661219, 45994669487254819820504, 120415608018950816800347, 315252154569597630580592, 825340855689842074941485, 2160770412499928594243920, 5656970381809943707790333, 14810140732929902529127138, 38773451816979763879591141, 101510214718009389109646346, 265757192337048403449347959, 695761362293135821238397594, 1821526894542359060265844887, 4768819321333941359559137132, 12484931069459465018411566575, 32685973887044453695675562660, 85572990591673896068615121473, 224032997887977234510169801828, 586526003072257807461894284081, 1535545011328796187875513050486, 4020109030914130756164644867449, 10524782081413596080618421551934, 27554237213326657485690619788427, 72137929558566376376453437813422, 188859551462372471643669693651915, 494440724828551038554555643142400, 1294462623023280644019997235775363, 3388947144241290893505436064183768, 8872378809700592036496310956776021, 23228189284860485215983496806144376, 60812189044880863611454179461657189, 159208377849782105618379041578827274, 416812944504465453243682945274824717, 1091230455663614254112669794245646962, 2856878422486377309094326437462116255, 7479404811795517673170309518140701890, 19581336012900175710416602116959989503, 51264603226905009458079496832739266708, 134212473667814852663821888381257810711, 351372817776539548533386168311034165516, 919905979661803792936336616551844685929, 2408345121208871830275623681344499892364, 6305129383964811697890534427481654991257, 16507043030685563263395979601100465081502, 43215999708091878092297404375819740253345, 113140956093590071013496233526358755678630, 296206868572678334948191296203256526782643
Generating function in Maple syntax:
(x^6-2*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{x^{6}-2 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:
(x**6 - 2*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)+x^6-2*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)+x^{6}-2 x^{5}+x^{4}-x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,(2*5^(1/2)-4)*(3/2-1/2*5^(1/2))^(-n)+(-2*5^(1/2)-4)*(3/2+1/2*5^(1/2))^(-n)+1-n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \left(2 \sqrt{5}-4\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-2 \sqrt{5}-4\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+1-n & \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) = 54
a(6) = 147
a(n+2) = -a(n)+3*a(n+1)+n, n >= 7
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) = 54
a \! \left(6\right) = 147
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17263/
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[25,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[24,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
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[19,x]
F[24,x] = F[20,x]
F[25,x] = F[26,x]+F[62,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[17,x]*F[4,x]
F[32,x] = F[26,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]+F[60,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[30,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[22,x]*F[4,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = 2*F[34,x]+F[42,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[38,x]
F[45,x] = F[41,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[34,x]+F[49,x]+F[59,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[4,x]+F[52,x]
F[52,x] = F[39,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = 2*F[34,x]+F[46,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[52,x]
F[58,x] = F[54,x]
F[59,x] = F[26,x]*F[4,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[26,x]+F[48,x]
F[62,x] = F[34,x]+F[35,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[26,x]+F[66,x]
F[66,x] = F[34,x]+F[49,x]+F[67,x]
F[67,x] = F[4,x]*F[65,x]
F[68,x] = F[69,x]+F[77,x]
F[69,x] = F[34,x]+F[35,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[61,x]+F[72,x]
F[72,x] = F[69,x]+F[73,x]
F[73,x] = 2*F[34,x]+F[55,x]+F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[47,x]+F[76,x]
F[76,x] = F[73,x]
F[77,x] = 2*F[34,x]+F[55,x]+F[74,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_{25}\! \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_{24}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \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_{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_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{62}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{34}\! \left(x \right) = 0
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_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{34}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \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_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{39}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{46}\! \left(x \right)+F_{55}\! \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_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{26}\! \left(x \right)+F_{48}\! \left(x \right)
F_{62}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{26}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{34}\! \left(x \right)+F_{49}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{77}\! \left(x \right)
F_{69}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \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_{61}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{55}\! \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_{47}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{55}\! \left(x \right)+F_{74}\! \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_25(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_24(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) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
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_19(x))
Eq(F_24(x), F_20(x))
Eq(F_25(x), F_26(x) + F_62(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_17(x)*F_4(x))
Eq(F_32(x), F_26(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_60(x))
Eq(F_34(x), 0)
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_30(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_22(x)*F_4(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), 2*F_34(x) + F_42(x) + F_46(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_38(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_34(x) + F_49(x) + F_59(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_4(x) + F_52(x))
Eq(F_52(x), F_39(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), 2*F_34(x) + F_46(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_52(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_26(x)*F_4(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_26(x) + F_48(x))
Eq(F_62(x), F_34(x) + F_35(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_26(x) + F_66(x))
Eq(F_66(x), F_34(x) + F_49(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_65(x))
Eq(F_68(x), F_69(x) + F_77(x))
Eq(F_69(x), F_34(x) + F_35(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_61(x) + F_72(x))
Eq(F_72(x), F_69(x) + F_73(x))
Eq(F_73(x), 2*F_34(x) + F_55(x) + F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_47(x) + F_76(x))
Eq(F_76(x), F_73(x))
Eq(F_77(x), 2*F_34(x) + F_55(x) + F_74(x))
Pack JSON:
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Specification JSON:
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