0132_0321_1203_1320_3021
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 372, 946, 2387, 6002, 15079, 37919, 95561, 241541, 612612, 1559353, 3983436, 10210626, 26255106, 67702588, 175017497, 453417572, 1176846421, 3059295689, 7963291679, 20750887319, 54121597842, 141261584371, 368924957442, 963969783732, 2519763167904, 6588608833174, 17232104687471, 45078714199526, 117943908368467, 308628456853763, 807683764154501, 2113890259664441, 5532887503210416, 14482504507233853, 37909953094072392, 99237734048239542, 259783457841345486, 680071957545568384, 1780348851911647829, 4660803074375441144, 12201708527493786337, 31943601228478095821, 83627617414312769147, 218936225158460570987, 573174865611581308254, 1500575705302331373079, 3928526354597555429406, 10284950441194713310536, 26926216882595527608972, 70493479530210128360698, 184553771348072120422091, 483166915779682249322906, 1264945102493529641418751, 3311664572648422664751143, 8670040833231582256615937, 22698442074375635760949037, 59425253108093196034514076, 155577251533378189752313729, 407306367753146838828176004, 1066341579636587239516325994, 2791717817754292668434251482, 7308810748374902269503777796, 19134712139974149000092679377, 50095321022968038155967250892, 131151241484196999728518643629, 343358384245009124371654997009, 898923872291306889711873331991, 2353413153529272956697407667071, 6161315427736051562812412401002, 16130532803835594413734309788283, 42230282322639424076640187043706, 110560312822930596681207010650396, 289450653426069271834065201815736, 757791641939553166829242985196958, 1983924261210026397218343887740007, 5193981119023166907051491652279182, 13598019049920293178580222430259691, 35600075937650424518362729186121099, 93202208574438552516625813873936829, 244006549403644674033291893228534369, 638817438862782945029887193999710152, 1672445765617936298835810278351388421, 4378519854818804600808756686625445632, 11463113792416663549797549605485737822, 30010821509432815636087399243390959654, 78569350709575557523650317283080747464, 205697230566062435245592200785188937101, 538522340880910964795298201156262252256, 1409869791858793012226075474301351952057, 3691087034254761417897332843883705375893, 9663391310014174413180449259422394729971, 25299086893985321124444501254130353939411, 66233869368297293484496894973665645368246, 173402521203538340215219600270666994779423
Generating function in Maple syntax:
-(4*x^8-15*x^7+20*x^6-35*x^5+62*x^4-62*x^3+33*x^2-9*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{8}-15 x^{7}+20 x^{6}-35 x^{5}+62 x^{4}-62 x^{3}+33 x^{2}-9 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**8 + 15*x**7 - 20*x**6 + 35*x**5 - 62*x**4 + 62*x**3 - 33*x**2 + 9*x - 1)/((x - 1)**3*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^3*F(x)+4*x^8-15*x^7+20*x^6-35*x^5+62*x^4-62*x^3+33*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{8}-15 x^{7}+20 x^{6}-35 x^{5}+62 x^{4}-62 x^{3}+33 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-2*5^(1/2)+10)*(3/2-1/2*5^(1/2))^(-n)+1/20*(2*5^(1/2)+10)*(3/2+1/2*5^(1/2))^(-n)+1/20*(5*n-15)*2^n+1/2*n^2-5/2*n+4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-2 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(2 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(5 n -15\right) 2^{n}}{20}\\+\frac{n^{2}}{2}-\frac{5 n}{2}+4 & \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) = 144
a(7) = 372
a(8) = 946
a(n+4) = -1/2*n^2-4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+7/2*n-4, 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) = 54
a \! \left(6\right) = 144
a \! \left(7\right) = 372
a \! \left(8\right) = 946
a \! \left(n +4\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{7 n}{2}-4, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19422/
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[19,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[29,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[9,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[24,x]
F[28,x] = F[12,x]*F[23,x]
F[29,x] = F[15,x]+F[30,x]+F[58,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[29,x]+F[40,x]
F[40,x] = 2*F[15,x]+F[41,x]+F[45,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[33,x]
F[44,x] = F[40,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]+F[52,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[17,x]+F[51,x]
F[51,x] = F[48,x]
F[52,x] = 2*F[15,x]+F[53,x]+F[57,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[37,x]
F[56,x] = F[52,x]
F[57,x] = F[12,x]*F[47,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[71,x]
F[60,x] = F[20,x]+F[61,x]
F[61,x] = F[15,x]+F[62,x]+F[70,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[10,x]+F[37,x]
F[65,x] = F[61,x]+F[66,x]
F[66,x] = 2*F[15,x]+F[57,x]+F[67,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[55,x]+F[69,x]
F[69,x] = F[66,x]
F[70,x] = F[12,x]*F[60,x]
F[71,x] = F[48,x]+F[72,x]
F[72,x] = 2*F[15,x]+F[53,x]+F[73,x]
F[73,x] = F[12,x]*F[71,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_{19}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{58}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{37}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{71}\! \left(x \right)
F_{60}\! \left(x \right) = F_{20}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{70}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{55}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{71}\! \left(x \right) = F_{48}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \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_19(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_29(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_9(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_24(x))
Eq(F_28(x), F_12(x)*F_23(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_58(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_29(x) + F_40(x))
Eq(F_40(x), 2*F_15(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_33(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x) + F_52(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_17(x) + F_51(x))
Eq(F_51(x), F_48(x))
Eq(F_52(x), 2*F_15(x) + F_53(x) + F_57(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_37(x))
Eq(F_56(x), F_52(x))
Eq(F_57(x), F_12(x)*F_47(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_71(x))
Eq(F_60(x), F_20(x) + F_61(x))
Eq(F_61(x), F_15(x) + F_62(x) + F_70(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_10(x) + F_37(x))
Eq(F_65(x), F_61(x) + F_66(x))
Eq(F_66(x), 2*F_15(x) + F_57(x) + F_67(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_55(x) + F_69(x))
Eq(F_69(x), F_66(x))
Eq(F_70(x), F_12(x)*F_60(x))
Eq(F_71(x), F_48(x) + F_72(x))
Eq(F_72(x), 2*F_15(x) + F_53(x) + F_73(x))
Eq(F_73(x), F_12(x)*F_71(x))
Pack JSON:
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Specification JSON:
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