0132_0231_1023_1032_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 419, 1098, 2828, 7194, 18134, 45392, 112990, 279957, 690922, 1699317, 4166781, 10189392, 24856163, 60500271, 146961410, 356324969, 862484281, 2084373207, 5030023785, 12122166421, 29177409643, 70146507206, 168457818191, 404140107892, 968624656896, 2319458393193, 5549421766890, 13266600372606, 31691392814749, 75650231368372, 180460678081458, 430202785054176, 1024934185998792, 2440415060630950, 5807487688403396, 13812773576616147, 32836120961432656, 78020880262864319, 185296388323677711, 439874379911707646, 1043767951946269869, 2475713116997378153, 5869820162299914228, 13911822653996116619, 32959779063901562039, 78060528411558244597, 184812442687618893075, 437410034662319158627, 1034925530611353696777, 2447918940170665305852, 5788380136055004482061, 13683380918386055411222, 32337781293838709254698, 76403160414860040499631, 180468087396183061148300, 426168169517848568468944, 1006135457363854307494791, 2374815904106025687558222, 5604082838559698927741320, 13221574175065206452520406, 31186640823636180418078058, 73546755024682495400007596, 173408537665200776465639386, 408782155900719449945031297, 963452321727600863918539686, 2270322869841717765964496313, 5348922221824653700601956305, 12599934363820929451945075996, 29675334345650795711477058919, 69879568882370349548087915667, 164525764833647498560476671958, 387300896749429189715388519261, 911581873789293657671772955237, 2145246006859326217490344388675, 5047709529234504028100565064941, 11875419842243822296173668286817, 27934598353716081140182833167831, 65701632112224336650315481599394, 154508198727753448157231368376891, 363303745547269915385021846818536, 854146725696545353125305810803844, 2007893096703688628709192861845733, 4719493526646467780905252895541278, 11091698769808579758094683044226194, 26064519652794910880858308600015969, 61242295472510502572616785340668952, 143881290520702821950527199544135310, 337994270432505535505126296514392572, 793903087465682306133429573063161532, 1864574501577627420580487075258904162, 4378718415497190920724442990255123616, 10281825359278786772846569033741191327, 24140720692848518814524156638637014508, 56674525367081507213505465499907537891
Generating function in Maple syntax:
(x^9-4*x^8+13*x^7-28*x^6+47*x^5-58*x^4+50*x^3-27*x^2+8*x-1)/(x^3-2*x^2+3*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{x^{9}-4 x^{8}+13 x^{7}-28 x^{6}+47 x^{5}-58 x^{4}+50 x^{3}-27 x^{2}+8 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**9 - 4*x**8 + 13*x**7 - 28*x**6 + 47*x**5 - 58*x**4 + 50*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**3*(x**3 - 2*x**2 + 3*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)^2*(x-1)^3*F(x)-x^9+4*x^8-13*x^7+28*x^6-47*x^5+58*x^4-50*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-x^{9}+4 x^{8}-13 x^{7}+28 x^{6}-47 x^{5}+58 x^{4}-50 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/317400*(-345*(((1/3*n+146/115)*23^(1/2)+I*n-172/15*I)*3^(1/2)+(I*n+438/115*I)*23^(1/2)+n-172/15)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-2760*(((1/12*n-47/184)*23^(1/2)+I*n+41/24*I)*3^(1/2)+(-1/4*I*n+141/184*I)*23^(1/2)-n-41/24)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+13800*n+211600)*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(2760*(((-1/12*n+47/184)*23^(1/2)+I*n+41/24*I)*3^(1/2)+(-1/4*I*n+141/184*I)*23^(1/2)+n+41/24)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+345*(((-1/3*n-146/115)*23^(1/2)+I*n-172/15*I)*3^(1/2)+(I*n+438/115*I)*23^(1/2)-n+172/15)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n+211600)*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(460*(3^(1/2)*(n-141/46)*23^(1/2)-12*n-41/2)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+230*(3^(1/2)*(n+438/115)*23^(1/2)+3*n-172/5)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n+211600)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1/2*n^2+3/2*n-2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-345 \left(\left(\left(\frac{n}{3}+\frac{146}{115}\right) \sqrt{23}+\mathrm{I} n -\frac{172 \,\mathrm{I}}{15}\right) \sqrt{3}+\left(\mathrm{I} n +\frac{438 \,\mathrm{I}}{115}\right) \sqrt{23}+n -\frac{172}{15}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-2760 \left(\left(\left(\frac{n}{12}-\frac{47}{184}\right) \sqrt{23}+\mathrm{I} n +\frac{41 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}+\frac{141 \,\mathrm{I}}{184}\right) \sqrt{23}-n -\frac{41}{24}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+13800 n +211600\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(2760 \left(\left(\left(-\frac{n}{12}+\frac{47}{184}\right) \sqrt{23}+\mathrm{I} n +\frac{41 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}+\frac{141 \,\mathrm{I}}{184}\right) \sqrt{23}+n +\frac{41}{24}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+345 \left(\left(\left(-\frac{n}{3}-\frac{146}{115}\right) \sqrt{23}+\mathrm{I} n -\frac{172 \,\mathrm{I}}{15}\right) \sqrt{3}+\left(\mathrm{I} n +\frac{438 \,\mathrm{I}}{115}\right) \sqrt{23}-n +\frac{172}{15}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n +211600\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(460 \left(\sqrt{3}\, \left(n -\frac{141}{46}\right) \sqrt{23}-12 n -\frac{41}{2}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+230 \left(\sqrt{3}\, \left(n +\frac{438}{115}\right) \sqrt{23}+3 n -\frac{172}{5}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n +211600\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400}\\-\frac{n^{2}}{2}+\frac{3 n}{2}-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) = 19
a(5) = 56
a(6) = 156
a(7) = 419
a(8) = 1098
a(9) = 2828
a(n+6) = -1/2*n^2-a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5)-1/2*n-1, 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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 156
a \! \left(7\right) = 419
a \! \left(8\right) = 1098
a \! \left(9\right) = 2828
a \! \left(n +6\right) = -\frac{n^{2}}{2}-a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)-\frac{n}{2}-1, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18754/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[44,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]+F[4,x]
F[28,x] = F[15,x]+F[24,x]+F[29,x]
F[29,x] = F[4,x]*F[7,x]
F[30,x] = F[31,x]+F[43,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[21,x]+F[4,x]
F[35,x] = F[31,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[31,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[39,x]*F[4,x]
F[42,x] = F[36,x]
F[43,x] = F[37,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[24,x]+F[25,x]+F[48,x]
F[48,x] = F[4,x]*F[46,x]
F[49,x] = F[23,x]+F[50,x]
F[50,x] = F[37,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_{22}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \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_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{44}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{43}\! \left(x \right)
F_{31}\! \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_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \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_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{37}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(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_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_44(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x) + F_4(x))
Eq(F_28(x), F_15(x) + F_24(x) + F_29(x))
Eq(F_29(x), F_4(x)*F_7(x))
Eq(F_30(x), F_31(x) + F_43(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_21(x) + F_4(x))
Eq(F_35(x), F_31(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_31(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_39(x)*F_4(x))
Eq(F_42(x), F_36(x))
Eq(F_43(x), F_37(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_24(x) + F_25(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_46(x))
Eq(F_49(x), F_23(x) + F_50(x))
Eq(F_50(x), F_37(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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