0231_1302_2301_3012
Counting sequence:
1, 1, 2, 6, 20, 65, 203, 617, 1847, 5484, 16207, 47756, 140437, 412407, 1209892, 3547118, 10394555, 30450968, 89187600, 261183236, 764792169, 2239300661, 6556339913, 19195388343, 56198282784, 164529143145, 481679637902, 1410167741319, 4128394956681, 12086215091552, 35383309723188, 103587164100019, 303258506742678, 887809401031726, 2599119699835956, 7609089314765669, 22276091169555671, 65214658717571605, 190919997600039899, 558930210321933380, 1636303000664698403, 4790378785069646236, 14024131459240111281, 41056515359065457443, 120195495930671203756, 351879768950226636238, 1030149842398962695623, 3015827529414055175512, 8829022043241448970664, 25847509301522725075556, 75670185566136146147581, 221529158289271057137257, 648540340964635994698965, 1898642043167255621628011, 5558392253414580254066920, 16272537811485201022052293, 47638862952651860440165702, 139465723757705119181924667, 408294549800821736776026541, 1195307598882922489062641256, 3499337075673674999794304188, 10244526162496335576020481343, 29991485250877221455499505950, 87801931810151010931340286366, 257045596944194542973640815076, 752516915584837463583672032761, 2203039907985271504280203243347, 6449535865111897431678999334545, 18881415958260592106755471659055, 55276515402771466130590330149468, 161825424630532493860996508445287, 473753959815089969421703408063404, 1386944078489853248802510834818333, 4060364746309560281484674335758399, 11886969437890163331293226351483892, 34799839730100725111223533494805710, 101878687546754549712360261487103171, 298256171774007812111253462540642520, 873163427438735739659364674004592384, 2556240048552080553501596662287344644, 7483551166347884731051952347480155233, 21908559836181336968819374849235769069, 64138666713997656808486374423350755505, 187769921830073420587596798060048564815, 549708083289127194477776906290545945392, 1609304482252855629421848247921559384753, 4711338609217708293707398861737068818718, 13792735765970395850847829882029493772511, 40379088766338338829446384931556523974753, 118212284876981960158898058192181113818864, 346073790240807834341551247275160452307908, 1013152469020243282304907161218197363455163, 2966066643670307468401077613456302082532038, 8683343922757456748426559845684850554609582, 25421027488305477092352109326188134740338900, 74421633452469335478329704955742034194803357, 217873944248777298626750088771378993892927935, 637839474631207225016319214605444852986630429, 1867314592392328291652262111830540782604197491, 5466679196952178154027008927113845815062683444, 16004042149160730567616275454050530532050642267
Generating function in Maple syntax:
-(5*x^4-12*x^3+13*x^2-6*x+1)/(2*x-1)/(x^4-6*x^3+8*x^2-5*x+1)
Generating function in latex syntax:
-\frac{5 x^{4}-12 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{4}-6 x^{3}+8 x^{2}-5 x +1\right)}
Generating function in sympy syntax:
(-5*x**4 + 12*x**3 - 13*x**2 + 6*x - 1)/((2*x - 1)*(x**4 - 6*x**3 + 8*x**2 - 5*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^4-6*x^3+8*x^2-5*x+1)*F(x)+5*x^4-12*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+5 x^{4}-12 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
8576/5067*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^5-13*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1))-17222/1689*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^5-13*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1))+70177/5067*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^5-13*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1))-43177/5067*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^5-13*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1))+9122/5067*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^5-13*_Z^4+22*_Z^3-18*_Z^2+7*_Z-1))
Explicit closed form in latex syntax:
\frac{8576 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-13 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{5067}-\frac{17222 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-13 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{1689}+\frac{70177 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-13 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{5067}-\frac{43177 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-13 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{5067}+\frac{9122 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-13 \textit{\_Z}^{4}+22 \textit{\_Z}^{3}-18 \textit{\_Z}^{2}+7 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{5067}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = 2*a(n)-13*a(n+1)+22*a(n+2)-18*a(n+3)+7*a(n+4), n >= 5
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(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21470/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[35,x]+F[8,x]
F[8,x] = F[0,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = x
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[19,x]+F[20,x]+F[22,x]
F[19,x] = 0
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[11,x]+F[18,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[24,x]+F[28,x]
F[24,x] = F[15,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[11,x]+F[25,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]
F[30,x] = F[13,x]*F[31,x]
F[31,x] = F[15,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[13,x]*F[34,x]
F[34,x] = F[25,x]+F[32,x]
F[35,x] = F[17,x]*F[36,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[19,x]+F[39,x]+F[41,x]
F[39,x] = F[13,x]*F[40,x]
F[40,x] = F[15,x]+F[38,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[15,x]+F[38,x]
F[43,x] = 2*F[19,x]+F[44,x]+F[49,x]
F[44,x] = F[13,x]*F[45,x]
F[45,x] = F[25,x]+F[46,x]
F[46,x] = 2*F[19,x]+F[44,x]+F[47,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[25,x]+F[46,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = F[25,x]+F[43,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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{35}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \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_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{25}\! \left(x \right)+F_{43}\! \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_13(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_13(x)*F_7(x))
Eq(F_7(x), F_35(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_22(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_11(x) + F_18(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_28(x))
Eq(F_24(x), F_15(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_11(x) + F_25(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_13(x)*F_31(x))
Eq(F_31(x), F_15(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_13(x)*F_34(x))
Eq(F_34(x), F_25(x) + F_32(x))
Eq(F_35(x), F_17(x)*F_36(x))
Eq(F_36(x), F_24(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_19(x) + F_39(x) + F_41(x))
Eq(F_39(x), F_13(x)*F_40(x))
Eq(F_40(x), F_15(x) + F_38(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_15(x) + F_38(x))
Eq(F_43(x), 2*F_19(x) + F_44(x) + F_49(x))
Eq(F_44(x), F_13(x)*F_45(x))
Eq(F_45(x), F_25(x) + F_46(x))
Eq(F_46(x), 2*F_19(x) + F_44(x) + F_47(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_25(x) + F_46(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_25(x) + F_43(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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