0132_0213_1302_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 52, 125, 273, 557, 1085, 2053, 3826, 7099, 13218, 24821, 47119, 90457, 175435, 343149, 675672, 1337015, 2655048, 5285341, 10538845, 21037253, 42023721, 83984309, 167890862, 335686771, 671258494, 1342378597, 2684591835, 5368987313, 10737742807, 21475213405, 42950108788, 85899847791, 171799267524, 343598041613, 687195516681, 1374390385309, 2748780031957, 5497559224805, 10995117499434, 21990233926187, 43980466644890, 87960931934293, 175921862350951, 351843723006985, 703687444125603, 1407374886152141, 2814749769976144, 5629499537375527, 11258999071904896, 22517998140672189, 45035996277891957, 90071992551991925, 180143985099826113, 360287970195101077, 720575940385228390, 1441151880765029603, 2882303761524146166, 5764607523041859269, 11529215046076729523, 23058430092145876321, 46116860184283536559, 92233720368558182077, 184467440737106754540, 368934881474203135199, 737869762948395084412, 1475739525896778120685, 2951479051793543278753, 5902958103587072625741, 11805916207174130293485, 23611832414348244543173, 47223664828696471894626, 94447329657392925384859, 188894659314785831085202, 377789318629571641135541, 755578637259143259812799, 1511157274518286495667897, 3022314549036572965799675, 6044629098073145904402733, 12089258196146291779863112, 24178516392292583528949655, 48357032784585167025196728, 96714065569170334015669661, 193428131138340667994495629, 386856262276681335949925413, 773712524553362671858456921, 1547425049106725343673082229, 3094850098213450687299781662, 6189700196426901374550511955, 12379400392853802749049182574, 24758800785707605498043608357, 49517601571415210996029414795, 99035203142830421991997848593, 198070406285660843983931398791, 396140812571321687967795039005, 792281625142643375935518711908, 1584563250285286751870962298191
Generating function in Maple syntax:
-(x^8-4*x^7+9*x^6-19*x^5+30*x^4-33*x^3+21*x^2-7*x+1)/(2*x-1)/(x-1)^6
Generating function in latex syntax:
-\frac{x^{8}-4 x^{7}+9 x^{6}-19 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1}{\left(2 x -1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-x**8 + 4*x**7 - 9*x**6 + 19*x**5 - 30*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^6*F(x)+x^8-4*x^7+9*x^6-19*x^5+30*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{6} F \! \left(x \right)+x^{8}-4 x^{7}+9 x^{6}-19 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1-59/30*n+1/120*n^5-1/24*n^3+5/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 1-\frac{59 n}{30}+\frac{n^{5}}{120}-\frac{n^{3}}{24}+\frac{5 \,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) = 52
a(6) = 125
a(7) = 273
a(8) = 557
a(n+1) = 2*a(n)-1/120*(n+4)*(n^4-9*n^3+21*n^2-79*n+90), 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) = 52
a \! \left(6\right) = 125
a \! \left(7\right) = 273
a \! \left(8\right) = 557
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n +4\right) \left(n^{4}-9 n^{3}+21 n^{2}-79 n +90\right)}{120}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18527/
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[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[1,x]+F[14,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]+F[28,x]+F[38,x]
F[27,x] = 0
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[31,x]+F[6,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[34,x]
F[37,x] = F[26,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[48,x]
F[40,x] = F[21,x]+F[41,x]
F[41,x] = F[27,x]+F[42,x]+F[47,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[32,x]
F[46,x] = F[41,x]
F[47,x] = F[12,x]*F[40,x]
F[48,x] = F[49,x]+F[58,x]
F[49,x] = F[27,x]+F[50,x]+F[54,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[14,x]+F[53,x]
F[53,x] = F[32,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[21,x]
F[57,x] = F[49,x]
F[58,x] = F[59,x]
F[59,x] = F[12,x]*F[48,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{6}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{26}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{21}\! \left(x \right)
F_{57}\! \left(x \right) = F_{49}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_20(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_1(x) + F_14(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_38(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_37(x))
Eq(F_30(x), F_31(x) + F_6(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_34(x))
Eq(F_37(x), F_26(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_48(x))
Eq(F_40(x), F_21(x) + F_41(x))
Eq(F_41(x), F_27(x) + F_42(x) + F_47(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_32(x))
Eq(F_46(x), F_41(x))
Eq(F_47(x), F_12(x)*F_40(x))
Eq(F_48(x), F_49(x) + F_58(x))
Eq(F_49(x), F_27(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_14(x) + F_53(x))
Eq(F_53(x), F_32(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_21(x))
Eq(F_57(x), F_49(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x)*F_48(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, 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": [2, 1, 3, 0], "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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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