0132_0213_1302_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 327, 738, 1617, 3483, 7432, 15777, 33392, 70528, 148701, 312980, 657587, 1379145, 2887270, 6033947, 12588666, 26221462, 54534531, 113256534, 234893349, 486553863, 1006650132, 2080394853, 4294990612, 8858396988, 18253641977, 37580999272, 77309451687, 158913835733, 326417566226, 670014956415, 1374389600062, 2817498619250, 5772436127295, 11819750089162, 24189255911481, 49478023360947, 101155069877856, 206708186156649, 422212465213944, 862017116338488, 1759218604618837, 3588805953254268, 7318349394687707, 14918173765893793, 30399297484999422, 61924494876613667, 126100789566665282, 256705178760433038, 522417556775317051, 1062849512059802750, 2161727821138231437, 4395513236314026655, 8935141660703517420, 18158513697558325677, 36893488147419623196, 74939897799445609332, 152185638608104394481, 308982963234635622864, 627189298506125429839, 1272825341085959780013, 2582544170319337990442, 5238875316933513470983, 10625324586456702592902, 21545797078092757201898, 43681889966544219195767, 88544371553805848782962, 179449926349046519206177, 363622219180962682602443, 736689171327664654549176, 1492267808586807888807985, 3022314549036572938115712, 6120186961799060198373360, 12391489651049949042237645, 25085210757003555376731492, 50774884423814425339319811, 102758694667243479851770617, 207935240973716218051296406, 420706185225890952799675179, 851083777008698938995169034, 1721510367131231944783714438, 3481706360490132023156008947, 7040783973435600313491096998, 14236310451781873161342366197, 28782105913385091391407189303, 58183181846412872920261507012, 117604303732111126115419591157, 237684487542793012780634766372, 480320735242727546660863243948, 970544990799738135520916570601, 1960897022228042355440216088120, 3961408125713216879677200976887, 8002044413940698096947942591365, 16162545152909924869082969627970, 32642002955876907088540111454607
Generating function in Maple syntax:
(x^7-11*x^6+29*x^5-48*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{x^{7}-11 x^{6}+29 x^{5}-48 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**7 - 11*x**6 + 29*x**5 - 48*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)-x^7+11*x^6-29*x^5+48*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)-x^{7}+11 x^{6}-29 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/24*n^2-3/4*n-1/4*n^3+1/24*n^4+1/4*2^n*n+3/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{n^{2}}{24}-\frac{3 n}{4}-\frac{n^{3}}{4}+\frac{n^{4}}{24}+\frac{2^{n} n}{4}+\frac{3 \,2^{n}}{4} & \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) = 138
a(7) = 327
a(n+2) = 1/24*n^4-7/12*n^3+35/24*n^2-4*a(n)+4*a(n+1)+1/12*n+1, 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) = 54
a \! \left(6\right) = 138
a \! \left(7\right) = 327
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{7 n^{3}}{12}+\frac{35 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{n}{12}+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21637/
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[14,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[14,x]*F[15,x]*F[18,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]*F[15,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[15,x]*F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[14,x]*F[23,x]
F[23,x] = F[24,x]+F[49,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[11,x]*F[18,x]
F[26,x] = F[27,x]*F[34,x]
F[27,x] = F[16,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[36,x]
F[29,x] = 0
F[30,x] = F[14,x]*F[31,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[14,x]*F[34,x]
F[34,x] = F[1,x]+F[32,x]
F[35,x] = F[30,x]
F[36,x] = F[14,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[16,x]+F[39,x]
F[39,x] = F[29,x]+F[30,x]+F[40,x]
F[40,x] = F[14,x]*F[38,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[14,x]*F[44,x]
F[44,x] = F[16,x]+F[42,x]
F[45,x] = 2*F[29,x]+F[46,x]+F[48,x]
F[46,x] = F[14,x]*F[47,x]
F[47,x] = F[39,x]+F[45,x]
F[48,x] = F[14,x]*F[41,x]
F[49,x] = F[50,x]
F[50,x] = F[51,x]+F[59,x]
F[51,x] = F[18,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[54,x]
F[54,x] = F[14,x]*F[55,x]
F[55,x] = F[11,x]+F[52,x]
F[56,x] = F[16,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[14,x]*F[15,x]*F[34,x]
F[59,x] = F[15,x]*F[16,x]*F[34,x]*F[57,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{49}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{34}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{14}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right) F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{59}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{52}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{34}\! \left(x \right) F_{57}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_14(x)*F_15(x)*F_18(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)*F_15(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_15(x)*F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_14(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_49(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_11(x)*F_18(x))
Eq(F_26(x), F_27(x)*F_34(x))
Eq(F_27(x), F_16(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_36(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_14(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_14(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_32(x))
Eq(F_35(x), F_30(x))
Eq(F_36(x), F_14(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_16(x) + F_39(x))
Eq(F_39(x), F_29(x) + F_30(x) + F_40(x))
Eq(F_40(x), F_14(x)*F_38(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_14(x)*F_44(x))
Eq(F_44(x), F_16(x) + F_42(x))
Eq(F_45(x), 2*F_29(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_14(x)*F_47(x))
Eq(F_47(x), F_39(x) + F_45(x))
Eq(F_48(x), F_14(x)*F_41(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_51(x) + F_59(x))
Eq(F_51(x), F_18(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_14(x)*F_55(x))
Eq(F_55(x), F_11(x) + F_52(x))
Eq(F_56(x), F_16(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_14(x)*F_15(x)*F_34(x))
Eq(F_59(x), F_15(x)*F_16(x)*F_34(x)*F_57(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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