0132_0213_0321_1032_1230

Counting sequence:
1, 1, 2, 6, 19, 56, 161, 455, 1271, 3519, 9674, 26439, 71902, 194720, 525422, 1413326, 3791279, 10145736, 27093101, 72213135, 192152611, 510536879, 1354648694, 3590082719, 9504137162, 25136077680, 66420131402, 175370136486, 462697953499, 1219984894856, 3214786748201, 8466696554855, 22287475343711, 58642395536079, 154235215811474, 405500927052039, 1065738420198982, 2800093718407280, 7354772359601942, 19313061994419086, 50702320516275479, 133078551973668216, 349220547135871061, 916242165315489615, 2403511312304828971, 6303961574469004319, 16531718380336429214, 43347553307878780799, 113647005995211153362, 297921953322915657120, 780911895715769013842, 2046728305954018753926, 5363895208147775182819, 14056116262621910033816, 36831393653984649332081, 96503103791998030028615, 252834935981608119299591, 662380779764689677027999, 1735222572795786288645914, 4545496353201183291790599, 11906600485448117169349102, 31187182515002389484384000, 81686035881917847402680702, 213945980187327966229541006, 560333103615578474416585919, 1467490783587009116787386856, 3843195351936164618610558461, 10064644934730517884141426255, 26356894882064171066672979571, 69020900233588592526693168719, 180741682292763582967897098374, 473290760114952706496794926239, 1239339701466657613216571851802, 3245233164584396836660885771920, 8497578536299849380476099920922, 22250444752641160975695408047526, 58260859082288969372176126791979, 152549281523882428462892971525736, 399428386909337014486188788750201, 1045835831073742693257105395852135, 2738320411471097920277677402025711, 7169707965527578855822458817791279, 18772209914646901078675313069700194, 49150317199671677031421240435579719, 128686938956420497749509542343931862, 336930289634953104336167414854272080, 882151707151217759231033892841889702, 2309640176189621349420076673175463886, 6047047287362267585127522136315640999, 15832173962157345174222284164537804056, 41451097617574716770157245224461600101, 108525037203756866569745075671940999055, 284133473638540954638687144984713231611, 743898221314746202179030309831996367679, 1947616325156303133263440848804223049774, 5099083861458254365174080315716364810239, 13349956608677247778749413320909202616482, 34951561770795155771618174171275597538880, 91506601665610340953683751464010630235042, 239572764956061776725134379287206728135686, 627222609624528929910700626801603464345779

Generating function in Maple syntax:
(x^6-3*x^5-2*x^4+7*x^2-5*x+1)/(x-1)/(x^2-3*x+1)/(x^2+2*x-1)

Generating function in latex syntax:
\frac{x^{6}-3 x^{5}-2 x^{4}+7 x^{2}-5 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right)}

Generating function in sympy syntax:
(x**6 - 3*x**5 - 2*x**4 + 7*x**2 - 5*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(x**2 + 2*x - 1))

Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2-3*x+1)*(x^2+2*x-1)*F(x)+x^6-3*x^5-2*x^4+7*x^2-5*x+1 = 0

Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{6}-3 x^{5}-2 x^{4}+7 x^{2}-5 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,(3/2-1/2*5^(1/2))^(-n)+(3/2+1/2*5^(1/2))^(-n)+1/4*(3*2^(1/2)+1)*(-1-2^(1/2))^(-n)-1/2+1/4*(-3*2^(1/2)+1)*(2^(1/2)-1)^(-n))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\frac{\left(3 \sqrt{2}+1\right) \left(-1-\sqrt{2}\right)^{-n}}{4}-\frac{1}{2}+\\\frac{\left(-3 \sqrt{2}+1\right) \left(\sqrt{2}-1\right)^{-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) = 56
a(6) = 161
a(n+4) = a(n)-a(n+1)-6*a(n+2)+5*a(n+3)-1, 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) = 56
a \! \left(6\right) = 161
a \! \left(n +4\right) = a \! \left(n \right)-a \! \left(n +1\right)-6 a \! \left(n +2\right)+5 a \! \left(n +3\right)-1, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18241/
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[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[41,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[16,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[28,x]+F[29,x]+F[39,x]
F[28,x] = 0
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[12,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = x^2
F[34,x] = F[35,x]+F[37,x]
F[35,x] = F[28,x]+F[29,x]+F[36,x]
F[36,x] = F[12,x]*F[20,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[35,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[20,x]+F[35,x]
F[41,x] = F[28,x]+F[42,x]+F[73,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[28,x]+F[46,x]+F[49,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[10,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[12,x]+F[48,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[28,x]+F[42,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[20,x]+F[52,x]
F[55,x] = F[28,x]+F[56,x]+F[69,x]+F[71,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[63,x]
F[58,x] = F[59,x]+F[61,x]
F[59,x] = F[60,x]
F[60,x] = F[10,x]*F[12,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[48,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = 2*F[28,x]+F[56,x]+F[65,x]
F[65,x] = F[12,x]*F[52,x]
F[66,x] = F[67,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[35,x]+F[68,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[52,x]+F[68,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[54,x]+F[75,x]
F[75,x] = F[76,x]+F[78,x]
F[76,x] = F[28,x]+F[29,x]+F[77,x]
F[77,x] = F[12,x]*F[40,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[75,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_{14}\! \left(x \right)+F_{17}\! \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_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \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_{41}\! \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_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = x^{2}
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{41}\! \left(x \right) = F_{28}\! \left(x \right)+F_{42}\! \left(x \right)+F_{73}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{28}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \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_{12}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right)+F_{42}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{28}\! \left(x \right)+F_{56}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \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_{63}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{66}\! \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_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{35}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{52}\! \left(x \right)+F_{68}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{54}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \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_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_41(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_26(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_16(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_39(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_12(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), x**2)
Eq(F_34(x), F_35(x) + F_37(x))
Eq(F_35(x), F_28(x) + F_29(x) + F_36(x))
Eq(F_36(x), F_12(x)*F_20(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_35(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_20(x) + F_35(x))
Eq(F_41(x), F_28(x) + F_42(x) + F_73(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_28(x) + F_46(x) + F_49(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_10(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_12(x) + F_48(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_28(x) + F_42(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_20(x) + F_52(x))
Eq(F_55(x), F_28(x) + F_56(x) + F_69(x) + F_71(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_63(x))
Eq(F_58(x), F_59(x) + F_61(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_10(x)*F_12(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_48(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), 2*F_28(x) + F_56(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_52(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_35(x) + F_68(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_52(x) + F_68(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_54(x) + F_75(x))
Eq(F_75(x), F_76(x) + F_78(x))
Eq(F_76(x), F_28(x) + F_29(x) + F_77(x))
Eq(F_77(x), F_12(x)*F_40(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_75(x))
Pack JSON:
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Specification JSON:
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