0213_1230_1320_3012_3102
Counting sequence:
1, 1, 2, 6, 19, 49, 109, 230, 472, 947, 1861, 3587, 6793, 12664, 23286, 42307, 76071, 135557, 239685, 420938, 734900, 1276403, 2206809, 3800023, 6519969, 11150844, 19015794, 32343555, 54882427, 92927257, 157035421, 264891182, 446082448, 750059699, 1259385901, 2111777387, 3536740921, 5916429952, 9886662510, 16504498243, 27526060815, 45866867693, 76364140149, 127038531218, 211181409644, 350806205939, 582352622529, 966110104255, 1601779013457, 2654156684292, 4395519555306, 7275527668227, 12036482514403, 19903296907009, 32896501441741, 54347807098742, 89749039316296, 148149585946547, 244456095576085, 403215891373523, 664839667120105, 1095833448521608, 1805618685847014, 2974175594608963, 4897463310908919, 8062031396218709, 13267556228289189, 21828041636378330, 35902113397707812, 59035124579005427, 97048723054681641, 159500302256583847, 262076965012140033, 430521661592504844, 707070960629310306, 1161009350570271747, 1905969373572714571, 3128284514864586217, 5133448741532044285, 8422233900212985278, 13815378138656139520, 22657808179596601523, 37153077955891340029, 60910973913854029883, 99844031285750070169, 163635072393974901904, 268139150290100487774, 439314336488821721155, 719653647194044070367, 1178708257902733999037, 1930302339731768153301, 3160691306489360459042, 5174614789710977018588, 8470607948545044206579, 13864145734090576376721, 22688978530814882480527, 37126272108919275923121, 60742623331927237384212, 99369415977053409371610, 162539932537380621818883, 265837762279040902336435
Generating function in Maple syntax:
(3*x^10+5*x^9-7*x^8-11*x^7+3*x^6+4*x^5+3*x^4-2*x^3-5*x^2+4*x-1)/(x^2+x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{3 x^{10}+5 x^{9}-7 x^{8}-11 x^{7}+3 x^{6}+4 x^{5}+3 x^{4}-2 x^{3}-5 x^{2}+4 x -1}{\left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(3*x**10 + 5*x**9 - 7*x**8 - 11*x**7 + 3*x**6 + 4*x**5 + 3*x**4 - 2*x**3 - 5*x**2 + 4*x - 1)/((x - 1)**3*(x**2 + x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)^2*(x-1)^3*F(x)+3*x^10+5*x^9-7*x^8-11*x^7+3*x^6+4*x^5+3*x^4-2*x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+3 x^{10}+5 x^{9}-7 x^{8}-11 x^{7}+3 x^{6}+4 x^{5}+3 x^{4}-2 x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,1/50*((-40*n+263)*5^(1/2)+90*n-575)*(-1/2*5^(1/2)-1/2)^(-n)+1/50*((40*n-263)*5^(1/2)+90*n-575)*(1/2*5^(1/2)-1/2)^(-n)+2*n^2+10*n+8)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(-40 n +263\right) \sqrt{5}+90 n -575\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{50}+\\\frac{\left(\left(40 n -263\right) \sqrt{5}+90 n -575\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{50}+2 n^{2}+10 n +8 & \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) = 49
a(6) = 109
a(7) = 230
a(8) = 472
a(9) = 947
a(10) = 1861
a(n+4) = 2*n^2-a(n)-2*a(n+1)+a(n+2)+2*a(n+3)+2*n-20, n >= 11
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) = 49
a \! \left(6\right) = 109
a \! \left(7\right) = 230
a \! \left(8\right) = 472
a \! \left(9\right) = 947
a \! \left(10\right) = 1861
a \! \left(n +4\right) = 2 n^{2}-a \! \left(n \right)-2 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)+2 n -20, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20589/
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[24,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[21,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[11,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[57,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[42,x]
F[29,x] = F[30,x]+F[7,x]
F[30,x] = F[26,x]+F[31,x]+F[37,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[7,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[15,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[36,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[15,x]
F[42,x] = F[25,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[7,x]
F[53,x] = F[46,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[46,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[11,x]+F[46,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[41,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[33,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_{24}\! \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_{21}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{57}\! \left(x \right)
F_{26}\! \left(x \right) = 0
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_{42}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)+F_{37}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \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_{15}\! \left(x \right)
F_{42}\! \left(x \right) = F_{25}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \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_{46}\! \left(x \right) = F_{47}\! \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_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{7}\! \left(x \right)
F_{53}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{46}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{41}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{33}\! \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_24(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_21(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_11(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_57(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_42(x))
Eq(F_29(x), F_30(x) + F_7(x))
Eq(F_30(x), F_26(x) + F_31(x) + F_37(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_7(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_15(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_36(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_15(x))
Eq(F_42(x), F_25(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_7(x))
Eq(F_53(x), F_46(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_46(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_11(x) + F_46(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_41(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_33(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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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