0213_1032_1230_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 54, 143, 367, 933, 2375, 6078, 15647, 40486, 105160, 273892, 714676, 1867093, 4881604, 12769545, 33413709, 87449919, 228900877, 599195672, 1568593701, 4106435704, 10750470974, 28144584758, 73682648082, 192902331463, 505022682690, 1322163024563, 3461462034907, 9062216031561, 23725174654595, 62113289477922, 162614663919131, 425730653964538, 1114577219798884, 2918000878940920, 7639425212356384, 20000274426968809, 52361397532722352, 137083917304210317, 358890352977092121, 939587139357260739, 2459871061422067369, 6440026038966512348, 16860207045862416897, 44140595083063255468, 115561578178154812730, 302544139410671161898, 792070839987956114142, 2073668380446564599611, 5428934301179202543630, 14213134522811875307927, 37210469266804720514311, 97418273276871415644333, 255044350562626952960639, 667714778409095999187270, 1748099984661565027091159, 4576585175570589620524174, 11981655542042098355407552, 31368381450542590505060812, 82123488809564452740061516, 215002084978116432354770749, 562882766124729288544182364, 1473646213395981542137352913, 3858055874063069890947382437, 10100521408792992792643874823, 26443508352315527702002826245, 69230003648152974190322266208, 181246502592142397960940216477, 474509504128272606661432287136, 1242282009792672812084266790390, 3252336525249741606621212130782, 8514727565956545174870123791466, 22291846172619882862109757477103, 58360810951903085522670501059994, 152790586683089344761233696353595, 400010949097364901927573891068659, 1047242260609005285243363230567889, 2741715832729650831190934357415227, 7177905237579947009939733652170282, 18791999880010189877626978966365011, 49198094402450622103550209424683234, 128802283327341675592631367852709084, 337208755579574403314560618856223328, 882823983411381532150874931983760952, 2311263194654570189578105345085635153, 6050965600552329030823306714531516360, 15841633607002416893571721577757857493, 41473935220454921634811630409249367537, 108580172054362347986462848819746496011, 284266580942632122285096367610253678625, 744219570773534018804945384741034344612, 1948392131377969934026378368903132713721, 5100956823360375783106947434988666955348, 13354478338703157415023860231373454665074, 34962478192749096461526787267462586706770, 91532956239544131968848051874655781630134, 239636390525883299443871072668477124020579
Generating function in Maple syntax:
-(x^8-2*x^7-x^5+x^4+10*x^3-13*x^2+6*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{8}-2 x^{7}-x^{5}+x^{4}+10 x^{3}-13 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**8 + 2*x**7 + x**5 - x**4 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^2+x-1)*(x-1)^3*F(x)+x^8-2*x^7-x^5+x^4+10*x^3-13*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-2 x^{7}-x^{5}+x^{4}+10 x^{3}-13 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n)+1/10*(-7*5^(1/2)+15)*(-1/2*5^(1/2)-1/2)^(-n)+1/10*(7*5^(1/2)+15)*(1/2*5^(1/2)-1/2)^(-n)-1/2*n^2-5/2*n-2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}+\frac{\left(-7 \sqrt{5}+15\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\\\frac{\left(7 \sqrt{5}+15\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}-\frac{n^{2}}{2}-\frac{5 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) = 54
a(6) = 143
a(7) = 367
a(8) = 933
a(n+4) = -1/2*n^2+a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)-5/2*n+1, n >= 9
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) = 54
a \! \left(6\right) = 143
a \! \left(7\right) = 367
a \! \left(8\right) = 933
a \! \left(n +4\right) = -\frac{n^{2}}{2}+a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-\frac{5 n}{2}+1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20560/
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[16,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[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[7,x]
F[15,x] = F[11,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[18,x]+F[19,x]+F[55,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[35,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[18,x]+F[26,x]+F[30,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[21,x]+F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[7,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[11,x]
F[35,x] = F[36,x]+F[50,x]
F[36,x] = F[18,x]+F[19,x]+F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[22,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[42,x]
F[46,x] = F[39,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[39,x]
F[50,x] = 2*F[18,x]+F[51,x]+F[53,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[39,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[11,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[58,x]+F[7,x]
F[58,x] = F[40,x]
F[59,x] = F[11,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_{16}\! \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_{7}\! \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_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \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_{19}\! \left(x \right)+F_{55}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{35}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \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_{7}\! \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_{11}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{50}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \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_{7}\! \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_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{22}\! \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_{42}\! \left(x \right)
F_{46}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{39}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{39}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{11}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{7}\! \left(x \right)
F_{58}\! \left(x \right) = F_{40}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \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_16(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_7(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_15(x))
Eq(F_14(x), F_7(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_55(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_35(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_18(x) + F_26(x) + F_30(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_21(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_7(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_11(x))
Eq(F_35(x), F_36(x) + F_50(x))
Eq(F_36(x), F_18(x) + F_19(x) + F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_22(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_42(x))
Eq(F_46(x), F_39(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_39(x))
Eq(F_50(x), 2*F_18(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_39(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_11(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_58(x) + F_7(x))
Eq(F_58(x), F_40(x))
Eq(F_59(x), F_11(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, 0, 3, 2], "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, 0, 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, 0, 3, 2], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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