0231_1032_2013
Counting sequence:
1, 1, 2, 6, 21, 74, 257, 883, 3019, 10306, 35174, 120063, 409878, 1399367, 4777721, 16312213, 55693546, 190149958, 649212941, 2216551914, 7567781505, 25838021387, 88216521131, 301190040010, 1028327116510, 3510928386631, 11987059317734, 40926380506791, 139731403405089, 477072852620213, 1628828603675042, 5561168709441646, 18987017630362501, 64825733102472442, 221328897149048321, 755664122391163779, 2579998695266599947, 8808666536284356242, 30074668754604836246, 102681341945851529359, 350576030274197343478, 1196941437205086604231, 4086613688271950512649, 13952571878677625219925, 47637060138166593538266, 162643096795311115920470, 555298266904911270958205, 1895906874029022854806474, 6473030962306268896378561, 22100310101167029916871995, 75455178480055581934788587, 257620093717888267965486042, 879570017911441908011489934, 3003039884209991096033100119, 10253019501017080567866446918, 35005998235648340078976404647, 119517953940559199179651220401, 408059819290940116560275220085, 1393203369282642067881989422226, 4756693838548688038408687570686, 16240368615629468017873517951733, 55448086785420495994680718387802, 189311609910423047942979854256833, 646350266070851199782559245234227, 2206777844462558703244271764056907, 7534410845708532413411952267434914, 25724087693909012247159237188361158, 87827529084218984161813009318736031, 299861940949057912152933537750133206, 1023792705627793680288108145319987015, 3495446940613058896846565591741461529, 11934202351196648226810046260440594197, 40745915523560475113547054127590191818, 139115257391848604000568124258024553062, 474969198520273465775178388860582316781, 1621646279297396655099577306555753425066, 5536646720149039688847952447408586132481, 18903294322001365445192655174623174059115, 64539883847707382403074715801339986400299, 220352946746826798721913552854435448895466, 752332019291892430081504779815940726798142, 2568622183673916122882192013560663503594087, 8769824696111879631365758494623114793735590, 29942054417099686279698649951389166235455911, 102228568276174985856063082816328389762607937, 349030164270500570864855031362540759095373621, 1191663520529652311747293959817481333412028802, 4068593753577608105259465776544770481644759182, 13891047973251127797543275186543991983058157669, 47427004385849294979654169193086270672806327994, 161925921596894924323530126399256986743344858241, 552849677615881107334812167210855465237072105955, 1887546867269734580692188416044908274958110288011, 6444488113847176108099129329757922996723778910898, 22002858720849235271012140486941876644604042099446, 75122458655702588867850303288251661785356911088303, 256484117181111884929376932179122894218005532176310, 875691551413042361981807122139988251624866803028423, 2989797971289945678068474624201707213144398868226345, 10207808782333697988310284252526852340800402839257301, 34851639186754900597104187761703994926453394348428986
Generating function in Maple syntax:
-(2*x^4+2*x^3-7*x^2+5*x-1)/(2*x^2-4*x+1)/(x^3+x^2-2*x+1)
Generating function in latex syntax:
-\frac{2 x^{4}+2 x^{3}-7 x^{2}+5 x -1}{\left(2 x^{2}-4 x +1\right) \left(x^{3}+x^{2}-2 x +1\right)}
Generating function in sympy syntax:
(-2*x**4 - 2*x**3 + 7*x**2 - 5*x + 1)/((2*x**2 - 4*x + 1)*(x**3 + x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x^2-4*x+1)*(x^3+x^2-2*x+1)*F(x)+2*x^4+2*x^3-7*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}-4 x +1\right) \left(x^{3}+x^{2}-2 x +1\right) F \! \left(x \right)+2 x^{4}+2 x^{3}-7 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
-275/1244208*(1/14*2^(1/3)*((I+27/31*31^(1/2))*3^(1/2)-81/31*I*31^(1/2)-1)*(47+3*31^(1/2)*3^(1/2))^(2/3)+196/5+((I-6/31*31^(1/2))*3^(1/2)-18/31*I*31^(1/2)+1)*2^(2/3)*(47+3*31^(1/2)*3^(1/2))^(1/3))*((-31/462*2^(1/3)*((I+485/961*31^(1/2))*3^(1/2)+919/961*I*31^(1/2)-145/31)*(47+3*31^(1/2)*3^(1/2))^(2/3)+43/66*2^(2/3)*((I+95/1333*31^(1/2))*3^(1/2)-529/1333*I*31^(1/2)+25/43)*(47+3*31^(1/2)*3^(1/2))^(1/3)+350/1023*I*31^(1/2)-1582/165)*(47/1176*2^(1/3)*((I-3/47*31^(1/2))*3^(1/2)-9/47*I*31^(1/2)+1)*(47+3*31^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(188+12*31^(1/2)*3^(1/2))^(1/3)+1/12*(188+12*31^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+(-4/21*2^(1/3)*((I-351/1364*31^(1/2))*3^(1/2)-67/682*I*31^(1/2)+13/44)*(47+3*31^(1/2)*3^(1/2))^(2/3)+17/33*2^(2/3)*((I-156/527*31^(1/2))*3^(1/2)-61/527*I*31^(1/2)+26/17)*(47+3*31^(1/2)*3^(1/2))^(1/3)-350/1023*I*31^(1/2)-1582/165)*(1/588*2^(1/3)*(3*31^(1/2)*3^(1/2)-47)*(47+3*31^(1/2)*3^(1/2))^(2/3)-1/6*(188+12*31^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+(-68/231*(2^(1/3)-3/8*2^(5/6))*((I-1/17*31^(1/2))*3^(1/2)+3/17*I*31^(1/2)-1)*(47+3*31^(1/2)*3^(1/2))^(2/3)-((I-1/11*31^(1/2))*3^(1/2)-3/11*I*31^(1/2)+1)*(2^(1/6)-4/3*2^(2/3))*(47+3*31^(1/2)*3^(1/2))^(1/3)-5824/165+728/55*2^(1/2))*(1-1/2*2^(1/2))^(-n)+(-68/231*((I-1/17*31^(1/2))*3^(1/2)+3/17*I*31^(1/2)-1)*(2^(1/3)+3/8*2^(5/6))*(47+3*31^(1/2)*3^(1/2))^(2/3)+(2^(1/6)+4/3*2^(2/3))*((I-1/11*31^(1/2))*3^(1/2)-3/11*I*31^(1/2)+1)*(47+3*31^(1/2)*3^(1/2))^(1/3)-5824/165-728/55*2^(1/2))*(1+1/2*2^(1/2))^(-n)-644/55*(-47/1176*((I+3/47*31^(1/2))*3^(1/2)-9/47*I*31^(1/2)-1)*2^(1/3)*(47+3*31^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(188+12*31^(1/2)*3^(1/2))^(1/3)+1/12*(188+12*31^(1/2)*3^(1/2))^(1/3)-1/3)^(-n))
Explicit closed form in latex syntax:
-\frac{275 \left(\frac{2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{27 \sqrt{31}}{31}\right) \sqrt{3}-\frac{81 \,\mathrm{I} \sqrt{31}}{31}-1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{14}+\frac{196}{5}+\left(\left(\mathrm{I}-\frac{6 \sqrt{31}}{31}\right) \sqrt{3}-\frac{18 \,\mathrm{I} \sqrt{31}}{31}+1\right) 2^{\frac{2}{3}} \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\left(-\frac{31 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{485 \sqrt{31}}{961}\right) \sqrt{3}+\frac{919 \,\mathrm{I} \sqrt{31}}{961}-\frac{145}{31}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{462}+\frac{43 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{95 \sqrt{31}}{1333}\right) \sqrt{3}-\frac{529 \,\mathrm{I} \sqrt{31}}{1333}+\frac{25}{43}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{66}+\frac{350 \,\mathrm{I} \sqrt{31}}{1023}-\frac{1582}{165}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{31}}{47}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{1176}-\frac{\mathrm{I} \sqrt{3}\, \left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}+\left(-\frac{4 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{351 \sqrt{31}}{1364}\right) \sqrt{3}-\frac{67 \,\mathrm{I} \sqrt{31}}{682}+\frac{13}{44}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{21}+\frac{17 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{156 \sqrt{31}}{527}\right) \sqrt{3}-\frac{61 \,\mathrm{I} \sqrt{31}}{527}+\frac{26}{17}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{33}-\frac{350 \,\mathrm{I} \sqrt{31}}{1023}-\frac{1582}{165}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{31}\, \sqrt{3}-47\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{588}-\frac{\left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+\left(-\frac{68 \left(2^{\frac{1}{3}}-\frac{3 \,2^{\frac{5}{6}}}{8}\right) \left(\left(\mathrm{I}-\frac{\sqrt{31}}{17}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{31}}{17}-1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{231}-\left(\left(\mathrm{I}-\frac{\sqrt{31}}{11}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{31}}{11}+1\right) \left(2^{\frac{1}{6}}-\frac{4 \,2^{\frac{2}{3}}}{3}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{5824}{165}+\frac{728 \sqrt{2}}{55}\right) \left(1-\frac{\sqrt{2}}{2}\right)^{-n}+\left(-\frac{68 \left(\left(\mathrm{I}-\frac{\sqrt{31}}{17}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{31}}{17}-1\right) \left(2^{\frac{1}{3}}+\frac{3 \,2^{\frac{5}{6}}}{8}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{231}+\left(2^{\frac{1}{6}}+\frac{4 \,2^{\frac{2}{3}}}{3}\right) \left(\left(\mathrm{I}-\frac{\sqrt{31}}{11}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{31}}{11}+1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{5824}{165}-\frac{728 \sqrt{2}}{55}\right) \left(1+\frac{\sqrt{2}}{2}\right)^{-n}-\frac{644 \left(-\frac{47 \left(\left(\mathrm{I}+\frac{3 \sqrt{31}}{47}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{1176}+\frac{\mathrm{I} \sqrt{3}\, \left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}}{55}\right)}{1244208}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+5) = -2*a(n)+2*a(n+1)+7*a(n+2)-11*a(n+3)+6*a(n+4), n >= 5
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) = 21
a \! \left(n +5\right) = -2 a \! \left(n \right)+2 a \! \left(n +1\right)+7 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14760/
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[19,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[6,x]
F[10,x] = F[11,x]+F[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[12,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[35,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[28,x]
F[24,x] = F[15,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[7,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[21,x]+F[22,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[2,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[20,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[19,x]+F[37,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[29,x]
F[42,x] = F[39,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{20}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{29}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(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_6(x))
Eq(F_10(x), F_11(x) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_12(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_35(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_28(x))
Eq(F_24(x), F_15(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_2(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_20(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_19(x) + F_37(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_29(x))
Eq(F_42(x), F_39(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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