0231_0321_2103_3012
Counting sequence:
1, 1, 2, 6, 20, 58, 148, 364, 878, 2083, 4877, 11318, 26095, 59855, 136731, 311346, 707158, 1602892, 3627267, 8197481, 18506180, 41742304, 94087089, 211950648, 477239937, 1074171420, 2416994314, 5437104130, 12228405343, 27497840898, 61825461609, 138991131949, 312440056779, 702285207847, 1578459709815, 3547574536099, 7972813332805, 17917481585228, 40265241784532, 90484421490821, 203333652760982, 456917844294996, 1026742598176262, 2307175474849417, 5184370827925829, 11649533422149533, 26176922298308802, 58820221305745072, 132170064593111219, 296987535514957605, 667332468858945820, 1499496303608163183, 3369363097391703096, 7570937035874101693, 17011827323673035105, 38225387607142627257, 85891957987881116318, 192998014224484203439, 433663588302070361428, 974435052764156582500, 2189539026966927939650, 4919855675152746955048, 11054826648549511511102, 24839990773443921752786, 55814990829241738875340, 125415216244535344765121, 281805562143142173954447, 633211588069966208686788, 1422813960466358625525421, 3197034753254022586935469, 7183673362095126525712485, 16141570244990906627952728, 36269784116421077045955380, 81497474344539772595544538, 183123179534922703476875877, 411474080518777088623791495, 924573923651964920518542594, 2077498853907038012561863806, 4668097745211358441906420555, 10489120777985166159511595422, 23568841104465377303288985535, 52958706450466757647647758826, 118997135450674022501114191692, 267384140336562641287014291424, 600806717194568254753428047277, 1350000453108734347621157644528, 3033423508519632451303490661459, 6816040799752534064723807018515, 15315504740992936664302206219097, 34413626931539006585040845675285, 77326718095514720913097562694665, 173751558917169018964559613034193, 390416209983423796261255380299987, 877257262600401746939115568483914, 1971179179599661678761774947178926, 4429199417945914616157480320243707, 9952320764165205736045681057374068, 22362661787412997696938046806651794, 50248444959185349681079404945166392, 112907231011769004349725326591373463, 253700245324337270165032252186595795
Generating function in Maple syntax:
(x+1)*(x^8-5*x^7-7*x^5+10*x^4-5*x^3+6*x^2-4*x+1)/(x-1)/(x^2+1)/(x^3-x^2-2*x+1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{\left(x +1\right) \left(x^{8}-5 x^{7}-7 x^{5}+10 x^{4}-5 x^{3}+6 x^{2}-4 x +1\right)}{\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x + 1)*(x**8 - 5*x**7 - 7*x**5 + 10*x**4 - 5*x**3 + 6*x**2 - 4*x + 1)/((x - 1)*(x**2 + 1)*(x**3 - x**2 - 2*x + 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+1)*(x^3-x^2-2*x+1)*(x^3+x^2+x-1)*F(x)-(x+1)*(x^8-5*x^7-7*x^5+10*x^4-5*x^3+6*x^2-4*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)-\left(x +1\right) \left(x^{8}-5 x^{7}-7 x^{5}+10 x^{4}-5 x^{3}+6 x^{2}-4 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/336336*(((-65065*I+17563*11^(1/2))*3^(1/2)+52689*I*11^(1/2)-65065)*(17+3*11^(1/2)*3^(1/2))^(1/3)-232232+((-158158*I-25844*11^(1/2))*3^(1/2)+77532*I*11^(1/2)+158158)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/336336*(((65065*I+17563*11^(1/2))*3^(1/2)-52689*I*11^(1/2)-65065)*(17+3*11^(1/2)*3^(1/2))^(1/3)-232232+((158158*I-25844*11^(1/2))*3^(1/2)-77532*I*11^(1/2)+158158)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/336336*((1122*I*3^(1/2)+10230)*(-28+84*I*3^(1/2))^(2/3)+24948*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-34188*(-28+84*I*3^(1/2))^(1/3)+469392)*(1/168*(I*3^(1/2)+5)*(-28+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/336336*((4554*I*3^(1/2)-6798)*(-28+84*I*3^(1/2))^(2/3)+4620*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)+54516*(-28+84*I*3^(1/2))^(1/3)+469392)*(1/84*(I*3^(1/2)-2)*(-28+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/336336*((51688*11^(1/2)*3^(1/2)-316316)*(17+3*11^(1/2)*3^(1/2))^(2/3)-35126*11^(1/2)*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)+130130*(17+3*11^(1/2)*3^(1/2))^(1/3)-232232)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/336336*((-5676*I*3^(1/2)-3432)*(-28+84*I*3^(1/2))^(2/3)-29568*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-20328*(-28+84*I*3^(1/2))^(1/3)+469392)*(1/6*(-28+84*I*3^(1/2))^(1/3)+1/3-1/168*(-28+84*I*3^(1/2))^(2/3)-1/56*I*3^(1/2)*(-28+84*I*3^(1/2))^(2/3))^(-n)-8/13*cos(1/2*n*Pi)+15/26*sin(1/2*n*Pi)-3/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-65065 \,\mathrm{I}+17563 \sqrt{11}\right) \sqrt{3}+52689 \,\mathrm{I} \sqrt{11}-65065\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-232232+\left(\left(-158158 \,\mathrm{I}-25844 \sqrt{11}\right) \sqrt{3}+77532 \,\mathrm{I} \sqrt{11}+158158\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{336336}\\+\\\frac{\left(\left(\left(65065 \,\mathrm{I}+17563 \sqrt{11}\right) \sqrt{3}-52689 \,\mathrm{I} \sqrt{11}-65065\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-232232+\left(\left(158158 \,\mathrm{I}-25844 \sqrt{11}\right) \sqrt{3}-77532 \,\mathrm{I} \sqrt{11}+158158\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{336336}\\+\\\frac{\left(\left(1122 \,\mathrm{I} \sqrt{3}+10230\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+24948 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-34188 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+469392\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{336336}\\+\\\frac{\left(\left(4554 \,\mathrm{I} \sqrt{3}-6798\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+4620 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+54516 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+469392\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{336336}\\+\\\frac{\left(\left(51688 \sqrt{11}\, \sqrt{3}-316316\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-35126 \sqrt{11}\, \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+130130 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-232232\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{336336}\\+\\\frac{\left(\left(-5676 \,\mathrm{I} \sqrt{3}-3432\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-29568 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-20328 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+469392\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}\right)^{-n}}{336336}\\-\frac{8 \cos \left(\frac{n \pi}{2}\right)}{13}+\frac{15 \sin \left(\frac{n \pi}{2}\right)}{26}-\frac{3}{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 58
a(6) = 148
a(7) = 364
a(8) = 878
a(9) = 2083
a(n) = a(n+2)+3*a(n+3)+2*a(n+4)+a(n+6)-3*a(n+7)+a(n+8)+6, n >= 10
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) = 58
a \! \left(6\right) = 148
a \! \left(7\right) = 364
a \! \left(8\right) = 878
a \! \left(9\right) = 2083
a \! \left(n \right) = a \! \left(n +2\right)+3 a \! \left(n +3\right)+2 a \! \left(n +4\right)+a \! \left(n +6\right)-3 a \! \left(n +7\right)+a \! \left(n +8\right)+6, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15848/
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[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[20,x]+F[55,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,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[13,x]*F[4,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[28,x]+F[29,x]+F[53,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[35,x]
F[31,x] = F[24,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[4,x]
F[35,x] = F[36,x]+F[41,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[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[28,x]+F[40,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[4,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = x^2
F[50,x] = F[39,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[44,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[20,x]+F[44,x]
F[55,x] = F[28,x]+F[56,x]+F[95,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[70,x]
F[58,x] = F[59,x]+F[7,x]
F[59,x] = F[28,x]+F[60,x]+F[62,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[7,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[17,x]+F[4,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[67,x]
F[67,x] = x^2
F[68,x] = F[69,x]
F[69,x] = F[17,x]*F[4,x]
F[70,x] = F[71,x]+F[83,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[20,x]+F[71,x]
F[75,x] = F[76,x]+F[78,x]
F[76,x] = F[77,x]
F[77,x] = F[38,x]*F[4,x]
F[78,x] = F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[71,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[71,x]
F[83,x] = 2*F[28,x]+F[84,x]+F[86,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[71,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[90,x]
F[88,x] = F[44,x]+F[89,x]
F[89,x] = F[82,x]
F[90,x] = F[91,x]+F[93,x]
F[91,x] = F[92,x]
F[92,x] = F[39,x]*F[4,x]
F[93,x] = F[94,x]
F[94,x] = F[4,x]*F[81,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[19,x]+F[97,x]
F[97,x] = F[100,x]+F[98,x]
F[98,x] = F[28,x]+F[45,x]+F[99,x]
F[99,x] = F[4,x]*F[54,x]
F[100,x] = 2*F[28,x]+F[101,x]+F[105,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[104,x]
F[103,x] = F[17,x]
F[104,x] = F[81,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[71,x]+F[89,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_{14}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{55}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \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_{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_{13}\! \left(x \right) F_{4}\! \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_{53}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \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_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \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_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{28}\! \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_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = x^{2}
F_{50}\! \left(x \right) = F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right)+F_{44}\! \left(x \right)
F_{55}\! \left(x \right) = F_{28}\! \left(x \right)+F_{56}\! \left(x \right)+F_{95}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{7}\! \left(x \right)
F_{59}\! \left(x \right) = F_{28}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{7}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \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_{17}\! \left(x \right)+F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = x^{2}
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{83}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{20}\! \left(x \right)+F_{71}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{71}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{71}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{90}\! \left(x \right)
F_{88}\! \left(x \right) = F_{44}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{82}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{93}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{19}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{100}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{28}\! \left(x \right)+F_{45}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{104}\! \left(x \right)
F_{103}\! \left(x \right) = F_{17}\! \left(x \right)
F_{104}\! \left(x \right) = F_{81}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{71}\! \left(x \right)+F_{89}\! \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_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_20(x) + F_55(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(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_13(x)*F_4(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_53(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_35(x))
Eq(F_31(x), F_24(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_4(x))
Eq(F_35(x), F_36(x) + F_41(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_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_28(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_50(x))
Eq(F_47(x), F_4(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), x**2)
Eq(F_50(x), F_39(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_44(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_20(x) + F_44(x))
Eq(F_55(x), F_28(x) + F_56(x) + F_95(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_70(x))
Eq(F_58(x), F_59(x) + F_7(x))
Eq(F_59(x), F_28(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_7(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_17(x) + F_4(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), x**2)
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_17(x)*F_4(x))
Eq(F_70(x), F_71(x) + F_83(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_20(x) + F_71(x))
Eq(F_75(x), F_76(x) + F_78(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_38(x)*F_4(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_71(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_71(x))
Eq(F_83(x), 2*F_28(x) + F_84(x) + F_86(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_71(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_90(x))
Eq(F_88(x), F_44(x) + F_89(x))
Eq(F_89(x), F_82(x))
Eq(F_90(x), F_91(x) + F_93(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_39(x)*F_4(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_4(x)*F_81(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_19(x) + F_97(x))
Eq(F_97(x), F_100(x) + F_98(x))
Eq(F_98(x), F_28(x) + F_45(x) + F_99(x))
Eq(F_99(x), F_4(x)*F_54(x))
Eq(F_100(x), F_101(x) + F_105(x) + 2*F_28(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_104(x))
Eq(F_103(x), F_17(x))
Eq(F_104(x), F_81(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_71(x) + F_89(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], 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