0123_0132_0231_1320
Counting sequence:
1, 1, 2, 6, 20, 65, 198, 572, 1595, 4350, 11702, 31211, 82788, 218798, 576809, 1517928, 3989444, 10475189, 27485514, 72079928, 188951183, 495167178, 1297337162, 3398417591, 8901061800, 23311059770, 61044700973, 159848209572, 418550260040, 1095903234545, 2869360770942, 7512581732276, 19669189733123, 51496598080758, 134823825735614, 352981321578083, 924133023901644, 2419443519931814, 6334249075502609, 16583406785792448, 43416177440306348, 113665537851988493, 297581260749381522, 779079893663599376, 2039661718776301655, 5339911859735074050, 13980087054568455698, 36600375692249361647, 95821092798737764560, 250863008257080200690, 656768143078735372757, 1719441843191590985916, 4501558230920967719408, 11785234538421172438793, 30854148762042270127494, 80777218503105079002188, 211477520258071848993419, 553655369292708232204014, 1449488641663248376067654, 3794810663783427952894043, 9934943565859817596401588, 26010020466141589063881758, 68095118697256078050382457, 178275337355008901997539736, 466730896826535141762781460, 1221917360042125550931890405, 3199021197134899566315057498, 8375146259032689258577613672, 21926417635303400430546442655, 57404106757557976475319028410, 150285902858731457879925266522, 393453602261358254933486014631, 1030074904810787022458591259768, 2696771113941890243518404724778, 7060238440556658570248856829949, 18483944214811635191532633590868, 48391594218045346452957979589144, 126690838467658603064559176464289, 331680921241598860535155292373774, 868351925370474774129778185791588, 2273374855096499053031922235264403, 5951772640372369567321474460522598, 15581943066927304013643473027339342, 40794056562222931203030888383561171, 106800226623368267054293079647469340, 279606623315135424877536125607090710, 732019643336545117413690847270283873, 1916452306723514147034287516396716272, 5017337276892025763030673902305768700, 13135559524068620020740738591292390205, 34389341295545948056557550673114995362, 90032464363033451663663931031139775296, 235708051794482861963898277626478681703, 617091691022272044286958972260644964338, 1615567021276047091014834779980153592354, 4229609372813296868993257649329210566623, 11073261097178698796436362731306267606944, 28990173918752510081258679671187171244514, 75897260659138252569225374535450404098469, 198701608058781089870188840441554356985708, 520207563517442701528883939801993298719168
Generating function in Maple syntax:
(x^7-3*x^6-4*x^5+15*x^4-24*x^3+19*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(-1+x)^3
Generating function in latex syntax:
\frac{x^{7}-3 x^{6}-4 x^{5}+15 x^{4}-24 x^{3}+19 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(x**7 - 3*x**6 - 4*x**5 + 15*x**4 - 24*x**3 + 19*x**2 - 7*x + 1)/((x - 1)**3*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)*(-1+x)^3*F(x)+x^7-3*x^6-4*x^5+15*x^4-24*x^3+19*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{3} F \! \left(x \right)+x^{7}-3 x^{6}-4 x^{5}+15 x^{4}-24 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-6*5^(1/2)+30)*(3/2-1/2*5^(1/2))^(-n)+1/20*(6*5^(1/2)+30)*(3/2+1/2*5^(1/2))^(-n)-n^2+3*n-3/4*2^n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-6 \sqrt{5}+30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(6 \sqrt{5}+30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-n^{2}+3 n -\frac{3 \,2^{n}}{4}\\-3 & \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) = 65
a(6) = 198
a(7) = 572
a(n+3) = -n^2+2*a(n)-7*a(n+1)+5*a(n+2)+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) = 65
a \! \left(6\right) = 198
a \! \left(7\right) = 572
a \! \left(n +3\right) = -n^{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+3 n +1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14796/
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[17,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[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[133,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[28,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[13,x]+F[19,x]+F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]
F[28,x] = F[18,x]+F[29,x]
F[29,x] = F[130,x]+F[19,x]+F[30,x]+F[62,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[23,x]+F[33,x]
F[33,x] = F[19,x]+F[34,x]+F[35,x]+F[37,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[16,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[26,x]
F[39,x] = F[29,x]+F[40,x]
F[40,x] = F[19,x]+F[41,x]+F[45,x]+F[46,x]+F[54,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[33,x]
F[44,x] = F[40,x]
F[45,x] = 0
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[12,x]
F[53,x] = F[49,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[27,x]
F[61,x] = F[57,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[48,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[129,x]+F[19,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[71,x]
F[68,x] = F[4,x]+F[69,x]
F[69,x] = F[13,x]+F[19,x]+F[70,x]
F[70,x] = F[4,x]*F[7,x]
F[71,x] = F[65,x]+F[72,x]
F[72,x] = F[19,x]+F[62,x]+F[73,x]+F[85,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[78,x]
F[75,x] = F[69,x]+F[76,x]
F[76,x] = F[19,x]+F[34,x]+F[35,x]+F[77,x]
F[77,x] = F[27,x]*F[4,x]
F[78,x] = F[72,x]+F[79,x]
F[79,x] = F[19,x]+F[45,x]+F[46,x]+F[80,x]+F[84,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[76,x]
F[83,x] = F[79,x]
F[84,x] = F[4,x]*F[57,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[27,x]+F[7,x]
F[90,x] = F[86,x]+F[91,x]
F[91,x] = 2*F[19,x]+F[110,x]+F[92,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[94,x]+F[96,x]
F[94,x] = F[27,x]+F[95,x]
F[95,x] = F[35,x]
F[96,x] = F[91,x]+F[97,x]
F[97,x] = 2*F[19,x]+F[102,x]+F[103,x]+F[98,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[100,x]+F[101,x]
F[100,x] = F[95,x]
F[101,x] = F[97,x]
F[102,x] = 0
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]
F[105,x] = F[106,x]
F[106,x] = F[107,x]*F[4,x]
F[107,x] = F[108,x]+F[109,x]
F[108,x] = F[12,x]
F[109,x] = F[106,x]
F[110,x] = F[111,x]*F[4,x]
F[111,x] = F[105,x]+F[112,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]
F[114,x] = F[115,x]*F[4,x]
F[115,x] = F[116,x]+F[117,x]
F[116,x] = F[12,x]+F[4,x]
F[117,x] = F[113,x]+F[118,x]
F[118,x] = 2*F[19,x]+F[110,x]+F[119,x]
F[119,x] = F[120,x]*F[4,x]
F[120,x] = F[121,x]+F[123,x]
F[121,x] = F[12,x]+F[122,x]
F[122,x] = F[35,x]
F[123,x] = F[118,x]+F[124,x]
F[124,x] = 2*F[19,x]+F[102,x]+F[103,x]+F[125,x]
F[125,x] = F[126,x]*F[4,x]
F[126,x] = F[127,x]+F[128,x]
F[127,x] = F[122,x]
F[128,x] = F[124,x]
F[129,x] = F[2,x]*F[4,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[132,x]+F[56,x]
F[132,x] = F[49,x]+F[86,x]
F[133,x] = F[134,x]*F[4,x]
F[134,x] = F[132,x]+F[135,x]
F[135,x] = F[2,x]+F[65,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_{17}\! \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_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{133}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{130}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \left(x \right)+F_{62}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = 0
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \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_{12}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{27}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{48}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{129}\! \left(x \right)+F_{19}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{71}\! \left(x \right) = F_{65}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{19}\! \left(x \right)+F_{62}\! \left(x \right)+F_{73}\! \left(x \right)+F_{85}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{78}\! \left(x \right)
F_{75}\! \left(x \right) = F_{69}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{78}\! \left(x \right) = F_{72}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{76}\! \left(x \right)
F_{83}\! \left(x \right) = F_{79}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{27}\! \left(x \right)+F_{7}\! \left(x \right)
F_{90}\! \left(x \right) = F_{86}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{110}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{94}\! \left(x \right) = F_{27}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{35}\! \left(x \right)
F_{96}\! \left(x \right) = F_{91}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{102}\! \left(x \right)+F_{103}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{101}\! \left(x \right)
F_{100}\! \left(x \right) = F_{95}\! \left(x \right)
F_{101}\! \left(x \right) = F_{97}\! \left(x \right)
F_{102}\! \left(x \right) = 0
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{4}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{109}\! \left(x \right)
F_{108}\! \left(x \right) = F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{106}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{4}\! \left(x \right)
F_{111}\! \left(x \right) = F_{105}\! \left(x \right)+F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{4}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{117}\! \left(x \right)
F_{116}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{117}\! \left(x \right) = F_{113}\! \left(x \right)+F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{110}\! \left(x \right)+F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right) F_{4}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{123}\! \left(x \right)
F_{121}\! \left(x \right) = F_{12}\! \left(x \right)+F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{35}\! \left(x \right)
F_{123}\! \left(x \right) = F_{118}\! \left(x \right)+F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{102}\! \left(x \right)+F_{103}\! \left(x \right)+F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{128}\! \left(x \right)
F_{127}\! \left(x \right) = F_{122}\! \left(x \right)
F_{128}\! \left(x \right) = F_{124}\! \left(x \right)
F_{129}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{56}\! \left(x \right)
F_{132}\! \left(x \right) = F_{49}\! \left(x \right)+F_{86}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right) F_{4}\! \left(x \right)
F_{134}\! \left(x \right) = F_{132}\! \left(x \right)+F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{2}\! \left(x \right)+F_{65}\! \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_17(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))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_133(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_28(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_13(x) + F_19(x) + F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x))
Eq(F_28(x), F_18(x) + F_29(x))
Eq(F_29(x), F_130(x) + F_19(x) + F_30(x) + F_62(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_23(x) + F_33(x))
Eq(F_33(x), F_19(x) + F_34(x) + F_35(x) + F_37(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_16(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_29(x) + F_40(x))
Eq(F_40(x), F_19(x) + F_41(x) + F_45(x) + F_46(x) + F_54(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_33(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), 0)
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_12(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_27(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_48(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_129(x) + F_19(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_71(x))
Eq(F_68(x), F_4(x) + F_69(x))
Eq(F_69(x), F_13(x) + F_19(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_7(x))
Eq(F_71(x), F_65(x) + F_72(x))
Eq(F_72(x), F_19(x) + F_62(x) + F_73(x) + F_85(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_78(x))
Eq(F_75(x), F_69(x) + F_76(x))
Eq(F_76(x), F_19(x) + F_34(x) + F_35(x) + F_77(x))
Eq(F_77(x), F_27(x)*F_4(x))
Eq(F_78(x), F_72(x) + F_79(x))
Eq(F_79(x), F_19(x) + F_45(x) + F_46(x) + F_80(x) + F_84(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_76(x))
Eq(F_83(x), F_79(x))
Eq(F_84(x), F_4(x)*F_57(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_27(x) + F_7(x))
Eq(F_90(x), F_86(x) + F_91(x))
Eq(F_91(x), F_110(x) + 2*F_19(x) + F_92(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_96(x))
Eq(F_94(x), F_27(x) + F_95(x))
Eq(F_95(x), F_35(x))
Eq(F_96(x), F_91(x) + F_97(x))
Eq(F_97(x), F_102(x) + F_103(x) + 2*F_19(x) + F_98(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_101(x))
Eq(F_100(x), F_95(x))
Eq(F_101(x), F_97(x))
Eq(F_102(x), 0)
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_107(x)*F_4(x))
Eq(F_107(x), F_108(x) + F_109(x))
Eq(F_108(x), F_12(x))
Eq(F_109(x), F_106(x))
Eq(F_110(x), F_111(x)*F_4(x))
Eq(F_111(x), F_105(x) + F_112(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x))
Eq(F_114(x), F_115(x)*F_4(x))
Eq(F_115(x), F_116(x) + F_117(x))
Eq(F_116(x), F_12(x) + F_4(x))
Eq(F_117(x), F_113(x) + F_118(x))
Eq(F_118(x), F_110(x) + F_119(x) + 2*F_19(x))
Eq(F_119(x), F_120(x)*F_4(x))
Eq(F_120(x), F_121(x) + F_123(x))
Eq(F_121(x), F_12(x) + F_122(x))
Eq(F_122(x), F_35(x))
Eq(F_123(x), F_118(x) + F_124(x))
Eq(F_124(x), F_102(x) + F_103(x) + F_125(x) + 2*F_19(x))
Eq(F_125(x), F_126(x)*F_4(x))
Eq(F_126(x), F_127(x) + F_128(x))
Eq(F_127(x), F_122(x))
Eq(F_128(x), F_124(x))
Eq(F_129(x), F_2(x)*F_4(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_56(x))
Eq(F_132(x), F_49(x) + F_86(x))
Eq(F_133(x), F_134(x)*F_4(x))
Eq(F_134(x), F_132(x) + F_135(x))
Eq(F_135(x), F_2(x) + F_65(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], 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