0123_0132_2103_2130_3102
Counting sequence:
1, 1, 2, 6, 19, 49, 112, 263, 612, 1408, 3232, 7425, 17121, 39680, 92492, 216835, 511073, 1210329, 2877998, 6866617, 16427892, 39387950, 94597934, 227493093, 547631343, 1319277010, 3180015988, 7668412721, 18497668447, 44630128485, 107699154268, 259926001595, 627373098660, 1514363463052, 3655565957932, 8824573537833, 21303125112237, 51428093720564, 124154627367164, 299729321566735, 723599540454797, 1746904952664657, 4217369452743290, 10181575743657613, 24580405079785332, 59342189070587162, 143264449216023578, 345870521360257869, 835004533336093851, 2015877966560476390, 4866757726394824804, 11749388793318238493, 28365527509863419755, 68480430661908828189, 165326366687415411208, 399133126772123431343, 963592557575336980836, 2326318136649614304856, 5616228654121520790904, 13558775148325214708433, 32733778453495427791161, 79026331222018134940712, 190786439501999222956076, 460599207890269451299291, 1111984851375331045309817, 2684568904108575666338889, 6481122648676980070548614, 15646814183232168926843809, 37774750984709222796982260, 91196316101874472305953222, 220167383103777300460299974, 531531082168268085219159861, 1283229547205104615001499399, 3097990176186728430011387194, 7479209898926359237657106644, 18056409972954065729649079241, 43592029843028927765008790071, 105240469656009456094964315733, 254072969150056894827683562100, 613386407947829869430989964003, 1480845785031940484790754049700, 3575077977988835047980523092964, 8631001740971637130826529197956, 20837081459869094403805851655929, 50305164660605289102029639575941, 121447410780906307525249637397404, 293199986222130479911826248952540, 707847383224690882668029315633767, 1708894752670722898450935427729445, 4125636888564829134967511632077249, 9960168529798215825898902100016210, 24045973948157675855484169829294581, 58052116426107633887752425320812788, 140150206800363124971008132939740466, 338352530026817640525627643551293234, 816855266853971540783350800969669789, 1972063063734716299539165331328246835, 4760981394323330702844269364067815118, 11494025852381256330740973377120786116, 27749033099085642805059439611613863221, 66992092050552210609125704673186709955
Generating function in Maple syntax:
(4*x^13+20*x^12+21*x^11-29*x^10-42*x^9+27*x^8+34*x^7-13*x^6-18*x^5-3*x^4+9*x^3+2*x^2-4*x+1)/(x^2+2*x-1)/(x^2+x-1)^3
Generating function in latex syntax:
\frac{4 x^{13}+20 x^{12}+21 x^{11}-29 x^{10}-42 x^{9}+27 x^{8}+34 x^{7}-13 x^{6}-18 x^{5}-3 x^{4}+9 x^{3}+2 x^{2}-4 x +1}{\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{3}}
Generating function in sympy syntax:
(4*x**13 + 20*x**12 + 21*x**11 - 29*x**10 - 42*x**9 + 27*x**8 + 34*x**7 - 13*x**6 - 18*x**5 - 3*x**4 + 9*x**3 + 2*x**2 - 4*x + 1)/((x**2 + x - 1)**3*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+2*x-1)*(x^2+x-1)^3*F(x)+4*x^13+20*x^12+21*x^11-29*x^10-42*x^9+27*x^8+34*x^7-13*x^6-18*x^5-3*x^4+9*x^3+2*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{3} F \! \left(x \right)+4 x^{13}+20 x^{12}+21 x^{11}-29 x^{10}-42 x^{9}+27 x^{8}+34 x^{7}-13 x^{6}-18 x^{5}-3 x^{4}+9 x^{3}+2 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 5,49,1/500*((-5*n^2+105*n+402)*5^(1/2)+25*n^2+235*n+750)*(-1/2*5^(1/2)-1/2)^(-n)+1/500*((5*n^2-105*n-402)*5^(1/2)+25*n^2+235*n+750)*(1/2*5^(1/2)-1/2)^(-n)-1/4*2^(1/2)*((-1-2^(1/2))^(-n)-(2^(1/2)-1)^(-n)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 49 & n =5 \\ \frac{\left(\left(-5 n^{2}+105 n +402\right) \sqrt{5}+25 n^{2}+235 n +750\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{500}+\\\frac{\left(\left(5 n^{2}-105 n -402\right) \sqrt{5}+25 n^{2}+235 n +750\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{500}-\\\frac{\sqrt{2}\, \left(\left(-1-\sqrt{2}\right)^{-n}-\left(\sqrt{2}-1\right)^{-n}\right)}{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) = 49
a(6) = 112
a(7) = 263
a(8) = 612
a(9) = 1408
a(10) = 3232
a(11) = 7425
a(12) = 17121
a(13) = 39680
a(n+8) = -a(n)-5*a(n+1)-5*a(n+2)+8*a(n+3)+10*a(n+4)-8*a(n+5)-5*a(n+6)+5*a(n+7), n >= 14
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) = 112
a \! \left(7\right) = 263
a \! \left(8\right) = 612
a \! \left(9\right) = 1408
a \! \left(10\right) = 3232
a \! \left(11\right) = 7425
a \! \left(12\right) = 17121
a \! \left(13\right) = 39680
a \! \left(n +8\right) = -a \! \left(n \right)-5 a \! \left(n +1\right)-5 a \! \left(n +2\right)+8 a \! \left(n +3\right)+10 a \! \left(n +4\right)-8 a \! \left(n +5\right)-5 a \! \left(n +6\right)+5 a \! \left(n +7\right), \quad n \geq 14
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16651/
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[22,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[4,x]
F[22,x] = F[23,x]+F[37,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[1,x]+F[12,x]
F[27,x] = F[23,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[23,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[23,x]*F[4,x]
F[34,x] = F[28,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[28,x]*F[4,x]
F[37,x] = F[19,x]+F[38,x]+F[52,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]+F[7,x]
F[41,x] = F[19,x]+F[20,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[45,x]+F[51,x]
F[45,x] = F[19,x]+F[20,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[15,x]+F[48,x]
F[48,x] = F[45,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[45,x]
F[51,x] = F[50,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[59,x]
F[54,x] = F[23,x]+F[55,x]
F[55,x] = F[19,x]+F[33,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[18,x]+F[4,x]
F[59,x] = F[60,x]+F[69,x]
F[60,x] = F[19,x]+F[61,x]+F[64,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]+F[45,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[31,x]+F[66,x]
F[66,x] = F[60,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[60,x]
F[69,x] = F[19,x]+F[68,x]+F[70,x]+F[73,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[18,x]+F[51,x]
F[73,x] = 0
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_{22}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \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_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \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_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right)+F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{7}\! \left(x \right)
F_{41}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{51}\! \left(x \right) = F_{50}\! \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_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{23}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{19}\! \left(x \right)+F_{33}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{69}\! \left(x \right)
F_{60}\! \left(x \right) = F_{19}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \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_{63}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{31}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{60}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{69}\! \left(x \right) = F_{19}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)+F_{73}\! \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_{72}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)
F_{73}\! \left(x \right) = 0
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_22(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_4(x))
Eq(F_22(x), F_23(x) + F_37(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_1(x) + F_12(x))
Eq(F_27(x), F_23(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_31(x) + F_34(x))
Eq(F_31(x), F_23(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_23(x)*F_4(x))
Eq(F_34(x), F_28(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_28(x)*F_4(x))
Eq(F_37(x), F_19(x) + F_38(x) + F_52(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x) + F_7(x))
Eq(F_41(x), F_19(x) + F_20(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_45(x) + F_51(x))
Eq(F_45(x), F_19(x) + F_20(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_15(x) + F_48(x))
Eq(F_48(x), F_45(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_45(x))
Eq(F_51(x), F_50(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_59(x))
Eq(F_54(x), F_23(x) + F_55(x))
Eq(F_55(x), F_19(x) + F_33(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_18(x) + F_4(x))
Eq(F_59(x), F_60(x) + F_69(x))
Eq(F_60(x), F_19(x) + F_61(x) + F_64(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x) + F_45(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_31(x) + F_66(x))
Eq(F_66(x), F_60(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_60(x))
Eq(F_69(x), F_19(x) + F_68(x) + F_70(x) + F_73(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_18(x) + F_51(x))
Eq(F_73(x), 0)
Pack JSON:
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Specification JSON:
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