0123_0132_1032_1203_2013
Counting sequence:
1, 1, 2, 6, 19, 59, 183, 568, 1763, 5472, 16984, 52715, 163617, 507835, 1576220, 4892277, 15184666, 47130218, 146282931, 454033459, 1409230595, 4373974716, 13575957607, 42137103416, 130785284964, 405931812499, 1259932540913, 3910582907703, 12137680535608, 37672974147737, 116929505350914, 362926196588350, 1126451563912787, 3496284197089275, 10851778787856175, 33681787927481312, 104541647979533211, 324476722726455808, 1007111956106848736, 3125877516300079419, 9702109271626694065, 30113439770986930931, 93466196829260872212, 290100699759409310701, 900415539049214863034, 2794712813976905461314, 8674239141690125694643, 26923132964119591946963, 83564111706335681302203, 259366574260697169601252, 805022855746211100750719, 2498632678944968983554360, 7755264611095604120264332, 24070816689033023461543715, 74711082746044039368185505, 231888512849227722224820847, 719736355236716190136006256, 2233920148456192609776204273, 6933648958217805551553433666, 21520683229890132844796307254, 66795969838126591144165126035, 207321558472597578984048811771, 643485358647682869796942742567, 1997252045781175200534993353736, 6199077695816123180589028872979, 19240718446096052411564029361504, 59719407384069332435227081438248, 185357299848024120486270273187723, 575312617990168413870374848924673, 1785657261354574574046351628212267, 5542329083911747842625325157824524, 17202299869725411941746350322398245, 53392556870530810399285402595407002, 165719999695504179040481532944045530, 514362298956237949063190949154534835, 1596479453739244657588858250059011507, 4955158360913238151807056283121080051, 15379837381695952404484359798517774988, 47735991598827101871041937645612336471, 148163133157394543764932869219958089464, 459869236853879583699282967458392043380, 1427343702160465852968890840020788466611, 4430194239638792102671605389282323489297, 13750451955770255891714099135305362511271, 42678699569471233528111188245936876000424, 132466292948052492687005170127092951490569, 411149330799927733952729609516584216982978, 1276126691969254435386300016795689526949358, 3960846368855815798845905220514161532338643, 12293688437367375130490445271059068813998907, 38157192004071379826857635829972895968946079, 118432422381069955279418812710432849439176880, 367590955580577240968746883402357617131529547, 1140930058745803102733098286037045747363534720, 3541222598618479263478713670821570091529781040, 10991258751436015031404887895867067891720872667, 34114706313053848196947761973638249422526152721, 105885341537780023854321999591736318359108239203, 328647283364776086594370886671076022969045590276, 1020056556407382107980060421986866318329662923549, 3166055010759926347794503265552335273348097009850
Generating function in Maple syntax:
(x^3+x^2+2*x-1)/(x^3+3*x-1)
Generating function in latex syntax:
\frac{x^{3}+x^{2}+2 x -1}{x^{3}+3 x -1}
Generating function in sympy syntax:
(x**3 + x**2 + 2*x - 1)/(x**3 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^3+3*x-1)*F(x)-x^3-x^2-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+3 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-2 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n < 0,-1/24*((1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+4*I*3^(1/2)-4)^(-n)*(-2*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n)*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)*((4+4*5^(1/2))^(-2/3*n+1/3)+1/2*(4+4*5^(1/2))^(-2/3*n+2/3))*((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n*(1/16*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/16*(5^(1/2)-1)*2^(1/3)*(I*3^(1/2)-1)*(5^(1/2)+1)^(2/3))^n-2/5*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n)*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)*5^(1/2)*((4+4*5^(1/2))^(-2/3*n+1/3)-3/2*(4+4*5^(1/2))^(-2/3*n+2/3))*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*2^(1/3)*(I*3^(1/2)-1)*(5^(1/2)+1)^(2/3))^n*(1/2*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/2*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n-1/5*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n)*((5+5^(1/2))*(I*3^(1/2)-1)*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*(5^(1/2)-5/3)*(1+I*3^(1/2))*(4+4*5^(1/2))^(-2/3*n+2/3))*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*2^(1/3)*(I*3^(1/2)-1)*(5^(1/2)+1)^(2/3))^n+1/5*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n)*((I*5^(1/2)*3^(1/2)+5^(1/2)+5)*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*I*(4+4*5^(1/2))^(-2/3*n+2/3)*5^(1/2)*3^(1/2))*(1/8*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/8*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+(-1/32*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/8*I*3^(1/2)-1/8)^(-n)*((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n*(I*(4+4*5^(1/2))^(-2/3*n+1/3)*3^(1/2)-1/2*(I*3^(1/2)+3/5*5^(1/2)-1)*(4+4*5^(1/2))^(-2/3*n+2/3))),n = 0,1,0 < n,1/240*(4*5^(1/2)*(((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-2*(I*3^(1/2)-2^(1/3)*(5^(1/2)+1)^(2/3)+1)*(5^(1/2)-1))^(-n)*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n)*(4+4*5^(1/2))^(-2/3*n+1/3)-3/2*((8*I*3^(1/2)-8)*(4+4*5^(1/2))^(1/3)-8*(I*3^(1/2)-2^(1/3)*(5^(1/2)+1)^(2/3)+1)*(5^(1/2)-1))^(-n)*(-1/1024*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/256*I*3^(1/2)-1/256)^(-n)*(4+4*5^(1/2))^(-2/3*n+2/3))*(1/8*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/8*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+20*(-1/128*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/32*I*3^(1/2)-1/32)^(-n)*((8*I*3^(1/2)-8)*(4+4*5^(1/2))^(1/3)-8*(I*3^(1/2)-2^(1/3)*(5^(1/2)+1)^(2/3)+1)*(5^(1/2)-1))^(-n)*((4+4*5^(1/2))^(-2/3*n+1/3)+1/2*(4+4*5^(1/2))^(-2/3*n+2/3))*((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+2*((1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+4*I*3^(1/2)-4)^(-n)*((5+5^(1/2))*(I*3^(1/2)-1)*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*(5^(1/2)-5/3)*(1+I*3^(1/2))*(4+4*5^(1/2))^(-2/3*n+2/3))*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n))*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*2^(1/3)*(I*3^(1/2)-1)*(5^(1/2)+1)^(2/3))^n-1/120*((1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+4*I*3^(1/2)-4)^(-n)*(1/8*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/8*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n*((5+5^(1/2))*(1+I*3^(1/2))*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*(5^(1/2)-5/3)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(-2/3*n+2/3))*(-1/256*(5^(1/2)-1)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}-\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(-2 \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{16}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{16}\right)^{n}-\frac{2 \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \sqrt{5}\, \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{2}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right)^{n}}{5}-\frac{\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\left(5+\sqrt{5}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}}{5}+\frac{\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\left(\mathrm{I} \sqrt{5}\, \sqrt{3}+\sqrt{5}+5\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \,\mathrm{I} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \sqrt{5}\, \sqrt{3}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}}{5}+\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{32}-\frac{\mathrm{I} \sqrt{3}}{8}-\frac{1}{8}\right)^{-n} \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n} \left(\mathrm{I} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}} \sqrt{3}-\frac{\left(\mathrm{I} \sqrt{3}+\frac{3 \sqrt{5}}{5}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right)\right)}{24} & n <0 \\ 1 & n =0 \\ \frac{\left(4 \sqrt{5}\, \left(\left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}-2 \left(\mathrm{I} \sqrt{3}-2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}+1\right) \left(\sqrt{5}-1\right)\right)^{-n} \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(\left(8 \,\mathrm{I} \sqrt{3}-8\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}-8 \left(\mathrm{I} \sqrt{3}-2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}+1\right) \left(\sqrt{5}-1\right)\right)^{-n} \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{1024}-\frac{\mathrm{I} \sqrt{3}}{256}-\frac{1}{256}\right)^{-n} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}+20 \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{128}-\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(\left(8 \,\mathrm{I} \sqrt{3}-8\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}-8 \left(\mathrm{I} \sqrt{3}-2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}+1\right) \left(\sqrt{5}-1\right)\right)^{-n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n}+2 \left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(5+\sqrt{5}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}}{240}\\-\\\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n} \left(\left(5+\sqrt{5}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}}{120} & 0<n \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n) = -3*a(n+2)+a(n+3), n >= 4
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(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16336/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[41,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[18,x]
F[30,x] = F[31,x]+F[36,x]
F[31,x] = F[13,x]+F[21,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[2,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[20,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[20,x]*F[4,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[33,x]+F[43,x]
F[43,x] = F[20,x]+F[44,x]
F[44,x] = 2*F[13,x]+F[40,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[34,x]
F[48,x] = F[44,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_{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_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{20}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{34}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_41(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_18(x))
Eq(F_30(x), F_31(x) + F_36(x))
Eq(F_31(x), F_13(x) + F_21(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_2(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_20(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_20(x)*F_4(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_33(x) + F_43(x))
Eq(F_43(x), F_20(x) + F_44(x))
Eq(F_44(x), 2*F_13(x) + F_40(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_34(x))
Eq(F_48(x), F_44(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": 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