0132_0213_1230_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 127, 273, 546, 1044, 1948, 3602, 6673, 12471, 23589, 45183, 87532, 171158, 337066, 667224, 1325523, 2639697, 5265163, 10512701, 21003814, 41981448, 83931432, 167825358, 335606341, 671160539, 1342260193, 2684449707, 5368817808, 10737541866, 21474976534, 42949831028, 85899523687, 171798891093, 343597606311, 687195015369, 1374389810218, 2748779374652, 5497558476148, 10995116649546, 21990232964409, 43980465559743, 87960930713437, 175921860981143, 351843721474036, 703687442414334, 1407374884246338, 2814749767858512, 5629499535027643, 11258999069307161, 22517998137803779, 45035996274730773, 90071992548514542, 180143985096007728, 360287970190915456, 720575940380647814, 1441151880760024813, 2882303761518686307, 5764607523035911833, 11529215046070260291, 23058430092138849304, 46116860184275913938, 92233720368549924142, 184467440737097819628, 368934881474193479631, 737869762948384662429, 1475739525896766884383, 2951479051793531178017, 5902958103587059608178, 11805916207174116304356, 23611832414348229525324, 47223664828696455788418, 94447329657392908128097, 188894659314785812613063, 377789318629571621380501, 755578637259143238704559, 1511157274518286473133308, 3022314549036572941762662, 6044629098073145878784218, 12089258196146291752580936, 24178516392292583499918499, 48357032784585166994328033, 96714065569170333982871547, 193428131138340667959672813, 386856262276681335912979126, 773712524553362671819284824, 1547425049106725343631578328, 3094850098213450687255836222, 6189700196426901374504011413, 12379400392853802749000009451, 24758800785707605497991641169, 49517601571415210995974527963, 99035203142830421991939912352, 198070406285660843983870279098, 396140812571321687967730597446, 792281625142643375935450805604, 1584563250285286751870890779703
Generating function in Maple syntax:
-(2*x^8-4*x^7+3*x^6+6*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{2 x^{8}-4 x^{7}+3 x^{6}+6 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-2*x**8 + 4*x**7 - 3*x**6 - 6*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+2*x^8-4*x^7+3*x^6+6*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{8}-4 x^{7}+3 x^{6}+6 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,8-31/4*n+19/8*n^2-3/4*n^3+1/8*n^4+5/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 8-\frac{31 n}{4}+\frac{19 n^{2}}{8}-\frac{3 n^{3}}{4}+\frac{n^{4}}{8}+\frac{5 \,2^{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) = 53
a(6) = 127
a(7) = 273
a(8) = 546
a(n+1) = 2*a(n)-1/8*(-2+n)*(n-7)*(n^2-n+8), 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) = 53
a \! \left(6\right) = 127
a \! \left(7\right) = 273
a \! \left(8\right) = 546
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-2+n \right) \left(n -7\right) \left(n^{2}-n +8\right)}{8}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18490/
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[21,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[19,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[20,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[25,x]
F[28,x] = F[22,x]
F[29,x] = F[30,x]+F[31,x]+F[53,x]
F[30,x] = 0
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[52,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[30,x]+F[35,x]+F[40,x]
F[35,x] = F[12,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[39,x]
F[39,x] = F[26,x]+F[37,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[39,x]+F[42,x]
F[42,x] = F[43,x]+F[50,x]
F[43,x] = F[30,x]+F[44,x]+F[46,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[12,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[26,x]
F[49,x] = F[43,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[42,x]
F[52,x] = F[29,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[63,x]
F[55,x] = F[22,x]+F[56,x]
F[56,x] = F[30,x]+F[57,x]+F[62,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[61,x]
F[59,x] = F[10,x]+F[60,x]
F[60,x] = F[30,x]+F[35,x]+F[38,x]
F[61,x] = F[56,x]
F[62,x] = F[12,x]*F[55,x]
F[63,x] = F[64,x]+F[75,x]
F[64,x] = F[30,x]+F[65,x]+F[71,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[26,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[26,x]+F[68,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[22,x]
F[74,x] = F[64,x]
F[75,x] = 2*F[30,x]+F[76,x]+F[83,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[37,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[68,x]+F[82,x]
F[82,x] = F[80,x]
F[83,x] = F[12,x]*F[63,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_{21}\! \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_{19}\! \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_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{53}\! \left(x \right)
F_{30}\! \left(x \right) = 0
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_{52}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \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_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{43}\! \left(x \right) = F_{30}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{26}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{52}\! \left(x \right) = F_{29}\! \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_{63}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{30}\! \left(x \right)+F_{57}\! \left(x \right)+F_{62}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{61}\! \left(x \right) = F_{56}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{75}\! \left(x \right)
F_{64}\! \left(x \right) = F_{30}\! \left(x \right)+F_{65}\! \left(x \right)+F_{71}\! \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_{67}\! \left(x \right) = F_{26}\! \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_{26}\! \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_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{22}\! \left(x \right)
F_{74}\! \left(x \right) = F_{64}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{76}\! \left(x \right)+F_{83}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{37}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{68}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{80}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_21(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_19(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))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_25(x))
Eq(F_28(x), F_22(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_53(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_52(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_30(x) + F_35(x) + F_40(x))
Eq(F_35(x), F_12(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_39(x))
Eq(F_39(x), F_26(x) + F_37(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_39(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_50(x))
Eq(F_43(x), F_30(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_12(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_26(x))
Eq(F_49(x), F_43(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_42(x))
Eq(F_52(x), F_29(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_63(x))
Eq(F_55(x), F_22(x) + F_56(x))
Eq(F_56(x), F_30(x) + F_57(x) + F_62(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_61(x))
Eq(F_59(x), F_10(x) + F_60(x))
Eq(F_60(x), F_30(x) + F_35(x) + F_38(x))
Eq(F_61(x), F_56(x))
Eq(F_62(x), F_12(x)*F_55(x))
Eq(F_63(x), F_64(x) + F_75(x))
Eq(F_64(x), F_30(x) + F_65(x) + F_71(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_26(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_26(x) + F_68(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_22(x))
Eq(F_74(x), F_64(x))
Eq(F_75(x), 2*F_30(x) + F_76(x) + F_83(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_37(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_68(x) + F_82(x))
Eq(F_82(x), F_80(x))
Eq(F_83(x), F_12(x)*F_63(x))
Pack JSON:
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Specification JSON:
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