0213_0231_1032_1230_1320_2031
Counting sequence:
1, 1, 2, 6, 18, 50, 134, 352, 914, 2358, 6064, 15578, 40030, 102976, 265314, 684774, 1770576, 4585994, 11897294, 30909280, 80404562, 209386902, 545790064, 1423788986, 3716663998, 9707328640, 25365476034, 66305213382, 173374003344, 453447697898, 1186197338414, 3103533704608, 8121048332210, 21252631970166, 55622352063664, 145584359449754, 381068449259806, 997492139310784, 2611141680700194, 6835383146975910, 17893873888861392, 46843902057399242, 122633021919964814, 321045268097845216, 840482441408456978, 2200360274685670230, 5760512620741587184, 15081001665678647162, 39482131736480443774, 103364654671948818688, 270610319351366192514, 708463207157405941638, 1854772968933875642640, 4855842751795292296106, 12712728827804190444014, 33282289688421750589984, 87134029899270190748786, 228119574829407453131574, 597224235221790176855344, 1563552194087240584371290, 4093430437513689571168222, 10716735227343750081024832, 28056767318182216499833314, 73453550586301834922617446, 192303851582460406972880208, 503457937291632118798898570, 1318069824247698405815872334, 3450751258750301993005444192, 9034183389377513325059135954, 23651797765684105412179763478, 61921207583385049624076650864, 162111820262104560590404975418, 424414243610621713818171435070, 1111130891089998839026822823296, 2908978390109555509229018370114, 7615804198958437479876263654214, 19938434043844113271397012720016, 52199497602008248533439609547114, 136660058091604591761432195576494, 357780675312763979684399155637920, 936681965088825321295892466538802, 2452265214362430068485618310962422, 6420113666664785325273813953478064, 16808075762662335334657869230054426, 44004113574778576623536570510497438, 115204264867377180606010766674356160, 301608680836342685695384319908995234, 789621777254794614203460857146653222, 2067256650144657225804718546321362384, 5412148171593066387876301304602931018, 14169187861423634960927730279467827214, 37095415406178653288658643090580148448, 97117058343959211987641033571469423250, 254255759599083271829628781713496950486, 665650220399440211792331269610909524464, 1742694901490298640090251562928108691194, 4562434483851102381485625730241372437246, 11944608549617350090223888728832323899904, 31271391164099727540586283616129036827394, 81869564940859584793706882354903275724550, 214337303654794917283621071750480996655632
Generating function in Maple syntax:
(2*x^6-2*x^5-2*x^4-7*x^3+12*x^2-6*x+1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
\frac{2 x^{6}-2 x^{5}-2 x^{4}-7 x^{3}+12 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**6 - 2*x**5 - 2*x**4 - 7*x**3 + 12*x**2 - 6*x + 1)/((2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2*F(x)+2*x^6-2*x^5-2*x^4-7*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+2 x^{6}-2 x^{5}-2 x^{4}-7 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/40*(24*5^(1/2)-40)*(3/2-1/2*5^(1/2))^(-n)+1/40*(-24*5^(1/2)-40)*(3/2+1/2*5^(1/2))^(-n)+1/8*2^n*(-3+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(24 \sqrt{5}-40\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(-24 \sqrt{5}-40\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{2^{n} \left(-3+n \right)}{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) = 18
a(5) = 50
a(6) = 134
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3), n >= 7
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) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 134
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22531/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[15,x]*F[7,x]
F[7,x] = F[20,x]+F[8,x]
F[8,x] = F[19,x]+F[9,x]
F[9,x] = F[10,x]*F[16,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[13,x]*F[15,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]*F[15,x]*F[16,x]
F[19,x] = F[13,x]*F[2,x]
F[20,x] = F[21,x]+F[29,x]
F[21,x] = F[22,x]*F[25,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[15,x]*F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[15,x]*F[22,x]*F[27,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[25,x]
F[29,x] = F[17,x]*F[23,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{20}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{19}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \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_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{2}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_15(x)*F_7(x))
Eq(F_7(x), F_20(x) + F_8(x))
Eq(F_8(x), F_19(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_16(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_13(x)*F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_13(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)*F_15(x)*F_16(x))
Eq(F_19(x), F_13(x)*F_2(x))
Eq(F_20(x), F_21(x) + F_29(x))
Eq(F_21(x), F_22(x)*F_25(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_15(x)*F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_15(x)*F_22(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_17(x)*F_23(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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": [0, 2, 3, 1], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [0, 2, 3, 1], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2, 1, 3], "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, 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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, 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