0123_0231_1302_3012
Counting sequence:
1, 1, 2, 6, 20, 64, 195, 576, 1672, 4803, 13696, 38821, 109457, 307151, 858210, 2388664, 6625320, 18318986, 50509507, 138911878, 381155816, 1043642941, 2852106212, 7780588871, 21191067825, 57628924989, 156503232370, 424467039146, 1149849162892, 3111348018228, 8410046322611, 22710077676236, 61268195458392, 165147248232791, 444783759093240, 1196981919077289, 3218885458381345, 8650066946769931, 23229769393801202, 62344210779555228, 167219317567763456, 448258136247518206, 1200971819523249747, 3215963113043832786, 8607418163424605080, 23026487517957460689, 61572132168813866620, 164569976182854942091, 439677910184497291937, 1174204028590505141753, 3134614884441876440226, 8364942477079830907726, 22314437394976878179652, 59505742400043882610856, 158630683033554744714595, 422742613693627197759064, 1126238185797947412201320, 2999538719959069883558251, 7986437275105206208999472, 21258384232175525810331053, 56570489500852358452167921, 150499795381855534169833671, 400287255899705210698172482, 1064390338970759013223573376, 2829610501718077608228411160, 7520593021646492233930396914, 19983897388904948581713164195, 53090133766655983553191552286, 141011878950142342715658519432, 374462593579421436465192673253, 994201798235993801656348520916, 2639103399976525081561731331791, 7004164301789604947226051830737, 18585596606832395159650164839221, 49308190922931873226128458421650, 130793465273195997202413517008114, 346880106826130143737742996674620, 919816071882383737397029878711196, 2438667856923114953255358952538067, 6464517526516053473388869512512228, 17133775057410228635168704872817336, 45405148622441089585869788435859711, 120307803405307228415440835136778792, 318728318402936703385700697750686641, 844281189636477016624405025741976065, 2236117307195960643410497701579201539, 5921672864186829669493294277680901970, 15679705625381336335805020405415456356, 41512254899772178494242466561736144752, 109890687397363388645148842875353157478, 290865881374787556779562753227833290931, 769791550650979800210389026703116531306, 2037056478267623830063794465402915897272, 5389936413296327130100715175174860240761, 14259880641365191894385323338607952588780, 37722530620886316115376450870438631857939, 99779058671811124804560645083590558343745, 263895862636012095794434135783191255046641, 697880833510102906714311100663675227191362, 1845382333474464819259078529998052275977878, 4879200949379918391659093242303443128841076
Generating function in Maple syntax:
-(3*x^7-18*x^6+44*x^5-68*x^4+63*x^3-33*x^2+9*x-1)/(x^2-3*x+1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{3 x^{7}-18 x^{6}+44 x^{5}-68 x^{4}+63 x^{3}-33 x^{2}+9 x -1}{\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-3*x**7 + 18*x**6 - 44*x**5 + 68*x**4 - 63*x**3 + 33*x**2 - 9*x + 1)/((x - 1)**4*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*(x-1)^4*F(x)+3*x^7-18*x^6+44*x^5-68*x^4+63*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)^{2} \left(x -1\right)^{4} F \! \left(x \right)+3 x^{7}-18 x^{6}+44 x^{5}-68 x^{4}+63 x^{3}-33 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
1/150*((-15*n-192)*5^(1/2)+45*n+450)*(3/2-1/2*5^(1/2))^(-n)+1/150*((15*n+192)*5^(1/2)+45*n+450)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-3/2*n^2+13/3*n-5
Explicit closed form in latex syntax:
\frac{\left(\left(-15 n -192\right) \sqrt{5}+45 n +450\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{150}+\frac{\left(\left(15 n +192\right) \sqrt{5}+45 n +450\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{150}+\frac{n^{3}}{6}-\frac{3 n^{2}}{2}+\frac{13 n}{3}-5
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 195
a(7) = 576
a(n+4) = 1/6*n^3-1/2*n^2-a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-5/3*n+1, n >= 8
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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 195
a \! \left(7\right) = 576
a \! \left(n +4\right) = \frac{n^{3}}{6}-\frac{n^{2}}{2}-a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{5 n}{3}+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15019/
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[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[14,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[62,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[22,x]+F[35,x]
F[35,x] = 2*F[15,x]+F[36,x]+F[40,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[26,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[54,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[27,x]
F[48,x] = F[43,x]+F[49,x]
F[49,x] = 2*F[15,x]+F[40,x]+F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[47,x]
F[53,x] = F[49,x]
F[54,x] = F[55,x]
F[55,x] = 2*F[15,x]+F[56,x]+F[60,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[31,x]
F[59,x] = F[55,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[43,x]+F[55,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[2,x]+F[43,x]
F[65,x] = F[66,x]+F[76,x]
F[66,x] = F[15,x]+F[67,x]+F[74,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[46,x]+F[69,x]
F[69,x] = F[66,x]+F[70,x]
F[70,x] = 2*F[15,x]+F[40,x]+F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[52,x]+F[73,x]
F[73,x] = F[70,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[2,x]+F[66,x]
F[76,x] = 2*F[15,x]+F[56,x]+F[77,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[43,x]+F[76,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_{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_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{62}\! \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_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \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_{54}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \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_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{27}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \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_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{31}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{43}\! \left(x \right)+F_{55}\! \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_{2}\! \left(x \right)+F_{43}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{76}\! \left(x \right)
F_{66}\! \left(x \right) = F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{46}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{71}\! \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_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{70}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{2}\! \left(x \right)+F_{66}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{43}\! \left(x \right)+F_{76}\! \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_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) + F_14(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_62(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_22(x) + F_35(x))
Eq(F_35(x), 2*F_15(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_54(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_27(x))
Eq(F_48(x), F_43(x) + F_49(x))
Eq(F_49(x), 2*F_15(x) + F_40(x) + F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_47(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), 2*F_15(x) + F_56(x) + F_60(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_31(x))
Eq(F_59(x), F_55(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_43(x) + F_55(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_2(x) + F_43(x))
Eq(F_65(x), F_66(x) + F_76(x))
Eq(F_66(x), F_15(x) + F_67(x) + F_74(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_46(x) + F_69(x))
Eq(F_69(x), F_66(x) + F_70(x))
Eq(F_70(x), 2*F_15(x) + F_40(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_52(x) + F_73(x))
Eq(F_73(x), F_70(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_2(x) + F_66(x))
Eq(F_76(x), 2*F_15(x) + F_56(x) + F_77(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_43(x) + F_76(x))
Pack JSON:
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Specification JSON:
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