0132_0213_1230_1302_3012
Counting sequence:
1, 1, 2, 6, 19, 52, 131, 321, 785, 1937, 4836, 12207, 31092, 79748, 205622, 532244, 1381665, 3594350, 9365369, 24430987, 63788079, 166657483, 435638082, 1139172641, 2979725234, 7795715862, 20398883516, 53383914486, 139718912455, 365705078152, 957261075551, 2505808048677, 6559623506717, 17171984372693, 44954175078192, 117686234488419, 308095919997120, 806584315772600, 2111622619266194, 5528214744372392, 14472884048404989, 37890162318266738, 99197052819420437, 259699896092537551, 679900435567947099, 1779997011161974447, 4660081799555146926, 12200230791644797301, 31940575385064725150, 83621424985190145642, 218923558817459871416, 573148969967040218922, 1500522788092977340867, 3928418268346082399788, 10284729765038823589787, 26925766522998225855369, 70492560796477431509225, 184551897851583856321625, 483163096728746247898044, 1264937320275878275983447, 3311648719981787060142924, 8670008551436010735220268, 22698376357860483380528942, 59425119369215829320442236, 155576979443930880426462489, 407305814350869573111840182, 1066340454385271466693925361, 2791715530358145197480311891, 7308806099795586087187479495, 19134702695241491322323421907, 50095301838354699952045449306, 131151202524674322569478679817, 343358305145371841274642596042, 898923711730848583629014044158, 2353412827685989055146510828532, 6161314766604749489001783368094, 16130531462683522223149392889855, 42230279602556344416058568630800, 110560307307206568106189749520919, 289450642243505478287807708927277, 757791619272194110360736681067509, 1983924215270845351057288343660573, 5193981025935878957225689016270472, 13598018861327865578393568087218571, 35600075555629865940336050706458904, 93202207800726028643154779700596192, 244006547836876812912820261175431946, 638817435690561596141079655575418256, 1672445759196122347922967296682997365, 4378519841820434192972310410560171322, 11463113766110437722523332568625694221, 30010821456201393959016207837850494487, 78569350601874774125562507586725162867, 205697230348184988363305707037933173303, 538522340440204310861383532147032351302, 1409869790967476184024222819863830933469, 3691087032452320720833121015975287251510, 9663391306369678937743212725953928183250, 25299086886617102010972142706110028472944, 66233869353402398932388347350093199788882, 173402521173431638460726524677313372650619
Generating function in Maple syntax:
-(4*x^9-11*x^8+4*x^6+x^5+19*x^4-36*x^3+25*x^2-8*x+1)/(2*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{9}-11 x^{8}+4 x^{6}+x^{5}+19 x^{4}-36 x^{3}+25 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**9 + 11*x**8 - 4*x**6 - x**5 - 19*x**4 + 36*x**3 - 25*x**2 + 8*x - 1)/((x - 1)**3*(2*x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^2+x-1)*(x-1)^3*F(x)+4*x^9-11*x^8+4*x^6+x^5+19*x^4-36*x^3+25*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{9}-11 x^{8}+4 x^{6}+x^{5}+19 x^{4}-36 x^{3}+25 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/10*(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)+1/10*(-3*5^(1/2)+5)*(-1/2*5^(1/2)-1/2)^(-n)+1/10*(3*5^(1/2)+5)*(1/2*5^(1/2)-1/2)^(-n)+1/2*n^2-11/2*n+2^(-1+n)+4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \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}+\\\frac{\left(-3 \sqrt{5}+5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(3 \sqrt{5}+5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{n^{2}}{2}-\frac{11 n}{2}+\\2^{-1+n}+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) = 52
a(6) = 131
a(7) = 321
a(8) = 785
a(9) = 1937
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)-1/2*(-1+n)*(n-12), n >= 10
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) = 131
a \! \left(7\right) = 321
a \! \left(8\right) = 785
a \! \left(9\right) = 1937
a \! \left(n +5\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right)-\frac{\left(-1+n \right) \left(n -12\right)}{2}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18455/
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[24,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[23,x]
F[23,x] = F[14,x]
F[24,x] = F[25,x]+F[39,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[28,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]+F[34,x]+F[38,x]
F[33,x] = 0
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[29,x]
F[37,x] = F[32,x]
F[38,x] = F[12,x]*F[31,x]
F[39,x] = F[33,x]+F[40,x]+F[65,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[43,x]+F[6,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[10,x]
F[46,x] = F[39,x]+F[47,x]
F[47,x] = 2*F[33,x]+F[48,x]+F[52,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[43,x]
F[51,x] = F[47,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[10,x]+F[58,x]
F[58,x] = F[44,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = 2*F[33,x]+F[52,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[58,x]
F[64,x] = F[60,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[25,x]+F[54,x]
F[68,x] = F[69,x]
F[69,x] = F[33,x]+F[34,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[78,x]
F[72,x] = F[25,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[12,x]
F[77,x] = F[73,x]
F[78,x] = F[69,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_{24}\! \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_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{33}\! \left(x \right) = 0
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_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{29}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)+F_{65}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{6}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{33}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{43}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \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_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{44}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{33}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \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_{25}\! \left(x \right)+F_{54}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{78}\! \left(x \right)
F_{72}\! \left(x \right) = F_{25}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right)
F_{77}\! \left(x \right) = F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{69}\! \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_24(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_23(x))
Eq(F_23(x), F_14(x))
Eq(F_24(x), F_25(x) + F_39(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_28(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_34(x) + F_38(x))
Eq(F_33(x), 0)
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_29(x))
Eq(F_37(x), F_32(x))
Eq(F_38(x), F_12(x)*F_31(x))
Eq(F_39(x), F_33(x) + F_40(x) + F_65(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_43(x) + F_6(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_10(x))
Eq(F_46(x), F_39(x) + F_47(x))
Eq(F_47(x), 2*F_33(x) + F_48(x) + F_52(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_43(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_10(x) + F_58(x))
Eq(F_58(x), F_44(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), 2*F_33(x) + F_52(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_58(x))
Eq(F_64(x), F_60(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_25(x) + F_54(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_33(x) + F_34(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_78(x))
Eq(F_72(x), F_25(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_12(x))
Eq(F_77(x), F_73(x))
Eq(F_78(x), F_69(x))
Pack JSON:
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Specification JSON:
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