0123_0132_0231_1203_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 146, 363, 863, 1993, 4519, 10131, 22557, 50020, 110657, 244471, 539692, 1190918, 2627349, 5795616, 12783580, 28196164, 62189846, 137165444, 302529518, 667251667, 1471671908, 3245876836, 7159009241, 15789694721, 34825271068, 76809556657, 169408813837, 373642905099, 824095373801, 1817599569009, 4008842051347, 8841779485422, 19501158549515, 43011159160813, 94864097818298, 209229354198216, 461469867570247, 1017803832972734, 2244837020158610, 4951143907903422, 10920091648796608, 24085020317769978, 53121184543462700, 117162460735742549, 258409941789276886, 569941068122039602, 1257044596979846255, 2772499135748995323, 6114939339620057654, 13486923276219990503, 29746345688189006859, 65607630715998103549, 144702184708216231483, 319150715104621505471, 703909060925241151961, 1552520306558698574760, 3424191328222018696293, 7552291717369278587859, 16657103741297255795864, 36738398810816530335546, 81029089339002339308681, 178715282419301934465228, 394168963649420399320344, 869367016637843138006120, 1917449315694988210536698, 4229067595039396820455520, 9327502206716636778981562, 20572453729128261768566919, 45373975053295920357659224, 100075452313308477494372720, 220723358355745216757387989, 486820691764786353872513829, 1073716835842881185239482080, 2368157030041507587236437005, 5223134751847801528345475929, 11519986339538484241930525343, 25408129709118476071097582493, 56039394170084753670540739197, 123598774679707991583012105583, 272605679068534459237121899154, 601250752307153672144784646735, 1326100279294015335872581512105, 2924806237656565130982285040326, 6450863227620283934109354848348, 14227826734534583204091291333851, 31380459706725731539164867837258, 69211782641071747012439090656366, 152651392016678077228969472784450, 336683243740081885997103813548484, 742578270121235519006646717900214, 1637807932259149115242262908736408, 3612299108258380116481629631177577, 7967176486637995751969905980416490, 17572160905535140619182074869735454, 38756620919328661354845779370819595
Generating function in Maple syntax:
(x^7-2*x^6+4*x^5-7*x^4+9*x^3-10*x^2+5*x-1)/(x^3+2*x-1)/(-1+x)^4
Generating function in latex syntax:
\frac{x^{7}-2 x^{6}+4 x^{5}-7 x^{4}+9 x^{3}-10 x^{2}+5 x -1}{\left(x^{3}+2 x -1\right) \left(-1+x \right)^{4}}
Generating function in sympy syntax:
(x**7 - 2*x**6 + 4*x**5 - 7*x**4 + 9*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+2*x-1)*(-1+x)^4*F(x)-x^7+2*x^6-4*x^5+7*x^4-9*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+2 x -1\right) \left(-1+x \right)^{4} F \! \left(x \right)-x^{7}+2 x^{6}-4 x^{5}+7 x^{4}-9 x^{3}+10 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
-29/16+337/944*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^3+2*_Z-1))+179/944*Sum(_alpha^(-n),_alpha = RootOf(_Z^3+2*_Z-1))+587/944*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3+2*_Z-1))-1/12*n^3-7/8*n^2-1/6*n+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
-\frac{29}{16}+\frac{337 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{944}+\frac{179 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{944}+\frac{587 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{944}-\frac{n^{3}}{12}-\frac{7 n^{2}}{8}-\frac{n}{6}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 146
a(7) = 363
a(n) = -1/6*n^3-2*n^2-2*a(n+2)+a(n+3)-11/6*n-2, 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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 146
a \! \left(7\right) = 363
a \! \left(n \right) = -\frac{n^{3}}{6}-2 n^{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-\frac{11 n}{6}-2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16105/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[53,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[28,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[13,x]+F[19,x]+F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[7,x]
F[27,x] = F[23,x]
F[28,x] = F[29,x]+F[44,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[34,x]
F[32,x] = F[33,x]+F[7,x]
F[33,x] = F[13,x]
F[34,x] = F[29,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[33,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[33,x]
F[43,x] = F[40,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[23,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[23,x]
F[52,x] = F[49,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[2,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[2,x]*F[4,x]
F[58,x] = F[18,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \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_{53}\! \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_{28}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{44}\! \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_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)
F_{34}\! \left(x \right) = F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \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_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \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_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \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_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{23}\! \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_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{23}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{18}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_53(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_28(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_13(x) + F_19(x) + F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_7(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_29(x) + F_44(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_34(x))
Eq(F_32(x), F_33(x) + F_7(x))
Eq(F_33(x), F_13(x))
Eq(F_34(x), F_29(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_33(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_43(x))
Eq(F_42(x), F_33(x))
Eq(F_43(x), F_40(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) + F_48(x))
Eq(F_47(x), F_23(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_23(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_2(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_2(x)*F_4(x))
Eq(F_58(x), F_18(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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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