0231_0312_2103_2130

Counting sequence:
1, 1, 2, 6, 20, 62, 174, 455, 1138, 2770, 6631, 15710, 36977, 86670, 202594, 472726, 1101735, 2565657, 5971527, 13893513, 32316839, 75157096, 174766751, 406360441, 944797902, 2196589897, 5106777410, 11872343495, 27600701531, 64165223845, 149168276450, 346777776160, 806166356504, 1874118843290, 4356812999320, 10128386123371, 23545681076058, 54737132297012, 127248499039291, 295817040092578, 687691459165211, 1598689127510826, 3716502040124042, 8639820191505416, 20085147024597355, 46692305000712557, 108546444817073169, 252339877416822925, 586619057232090023, 1363723877237182610, 3170273419836729931, 7369991602997937965, 17133151912460098112, 39829746057853732005, 92592926124174081674, 215252438448442605441, 500401209606536253895, 1163291678777768226337, 2704325056751248238012, 6286793024218188568988, 14615020641025854970824, 33975800936401897578196, 78984154559477650439516, 183615882459769951774551, 426855139217976871740212, 992317807328203722468192, 2306859026063983155771355, 5362796602843410034545740, 12466989563875914435061191, 28982234512240244452787710, 67375521012193099504207100, 156629083576590847084449740, 364118443218623494721483739, 846472683516494920565860175, 1967810247691438558428922725, 4574603819264172299032490361, 10634663645932967610051301993, 24722593546973052669919620342, 57473057168335283877336333251, 133608648057021658570270416503, 310602423381414294082755141796, 722063031198610627109807216601, 1678592894890147563315625009132, 3902255045654834119516205250301, 9071642378383138860314805686815, 21089009938730415296221453817627, 49026000105080642679832410312464, 113971622816165596090644406217120, 264951878177068118384046584051658, 615938389003957365609515359465400, 1431883033473907216285486058146156, 3328724200590884356114694790623945, 7738344923828811081610237107741872, 17989469403778596149207638043881540, 41820442564269090121065574443235429, 97220733809078953027260954329889256, 226010785702477378350277061863729055, 525411332053543486359661831071494592, 1221433158564754926009397013230411672, 2839487597289655621505136108522317688, 6601007806793001207305972488701459309

Generating function in Maple syntax:
-(3*x^7-6*x^6+x^5+6*x^4-14*x^3+14*x^2-6*x+1)/(x^2+x-1)/(x^3-2*x^2+3*x-1)/(x-1)^3

Generating function in latex syntax:
-\frac{3 x^{7}-6 x^{6}+x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(x^{2}+x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-3*x**7 + 6*x**6 - x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(x**2 + x - 1)*(x**3 - 2*x**2 + 3*x - 1))

Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)+3*x^7-6*x^6+x^5+6*x^4-14*x^3+14*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{7}-6 x^{6}+x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0

Explicit closed form in Maple syntax:
1/69000*(1265*2^(2/3)*((423/253*I-141/253*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600-3197*((-693/3197*I-231/3197*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)+1/69000*(-1265*2^(2/3)*((423/253*I+141/253*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600+3197*((-693/3197*I+231/3197*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-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)+1/69000*((1410*3^(1/2)*2^(2/3)*23^(1/2)+2530*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600+(-462*23^(1/2)*3^(1/2)*2^(1/3)+6394*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)+1/69000*(41400*5^(1/2)-96600)*(-1/2*5^(1/2)-1/2)^(-n)+1/69000*(-41400*5^(1/2)-96600)*(1/2*5^(1/2)-1/2)^(-n)-1/2*n^2-1/2*n+4

Explicit closed form in latex syntax:
\frac{\left(1265 \,2^{\frac{2}{3}} \left(\left(\frac{423 \,\mathrm{I}}{253}-\frac{141 \sqrt{3}}{253}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600-3197 \left(\left(-\frac{693 \,\mathrm{I}}{3197}-\frac{231 \sqrt{3}}{3197}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{69000}+\frac{\left(-1265 \,2^{\frac{2}{3}} \left(\left(\frac{423 \,\mathrm{I}}{253}+\frac{141 \sqrt{3}}{253}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600+3197 \left(\left(-\frac{693 \,\mathrm{I}}{3197}+\frac{231 \sqrt{3}}{3197}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{69000}+\frac{\left(\left(1410 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}+2530 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600+\left(-462 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+6394 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{69000}+\frac{\left(41400 \sqrt{5}-96600\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{69000}+\frac{\left(-41400 \sqrt{5}-96600\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{69000}-\frac{n^{2}}{2}-\frac{n}{2}+4

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 174
a(7) = 455
a(n+1) = a(n)+4*a(n+3)-4*a(n+4)+a(n+5)+1/2*(n+4)*(n-3), 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) = 62
a \! \left(6\right) = 174
a \! \left(7\right) = 455
a \! \left(n +1\right) = a \! \left(n \right)+4 a \! \left(n +3\right)-4 a \! \left(n +4\right)+a \! \left(n +5\right)+\frac{\left(n +4\right) \left(n -3\right)}{2}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15795/
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[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[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[32,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[11,x]+F[29,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[17,x]+F[34,x]
F[34,x] = F[35,x]+F[44,x]
F[35,x] = F[19,x]+F[36,x]+F[42,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[19,x]+F[30,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[38,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[2,x]+F[35,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[47,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_{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_{11}\! \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_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{32}\! \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_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{44}\! \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_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \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_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_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_32(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_11(x) + F_29(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_17(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_44(x))
Eq(F_35(x), F_19(x) + F_36(x) + F_42(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_19(x) + F_30(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_2(x) + F_35(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_14(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_47(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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