0132_1230_2301_3012
Counting sequence:
1, 1, 2, 6, 20, 63, 180, 468, 1126, 2553, 5542, 11666, 24048, 48899, 98592, 197808, 395826, 791101, 1580418, 3157198, 6308108, 12606279, 25197740, 50374284, 100719198, 201398721, 402744958, 805421706, 1610756104, 3221401931, 6442666200, 12885162344, 25770116586, 51539980677, 103079657370, 206158951366, 412317471204, 824634433039, 1649268268196, 3298535838276, 6597070865366, 13194140792457, 26388280504278, 52776559768962, 105553118121376, 211106234629779, 422212467429136, 844424932787744, 1688849863240482, 3377699723855309, 6755399444766258, 13510798886239422, 27021597768804924, 54043195533520855, 108086391062501148, 216172782119971324, 432345564234379982, 864691128462621777, 1729382256918483374, 3458764513829535354, 6917529027650916024, 13835058055292899035, 27670116110576028616, 55340232221141390040, 110680464442271150554, 221360928884529641237, 442721857769045520714, 885443715538076102582, 1770887431076136010260, 3541774862152254486687, 7083549724304490013716, 14167099448608959550900, 28334198897217897013062, 56668397794435770225433, 113336795588871514833926, 226673591177743002125682, 453347182355485974670160, 906694364710971917601315, 1813388729421943801181952, 3626777458843887565932432, 7253554917687775092888082, 14507109835375550144114013, 29014219670751100243734754, 58028439341502200439993518, 116056878683004400829370732, 232113757366008801604821095, 464227514732017603152247692, 928455029464035206243450220, 1856910058928070412422021438, 3713820117856140824775140065, 7427640235712281649477156574, 14855280471424563298876764778, 29710560942849126597671345000, 59421121885698253195255650411, 118842243771396506390419179704, 237684487542793012780740922440, 475368975085586025561378849738, 950737950171172051122648895653, 1901475900342344102245182919930, 3802951800684688204490244633510, 7605903601369376408980361449540
Generating function in Maple syntax:
(2*x^7-3*x^6+16*x^5-31*x^4+33*x^3-21*x^2+7*x-1)/(2*x-1)/(x-1)^6
Generating function in latex syntax:
\frac{2 x^{7}-3 x^{6}+16 x^{5}-31 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:
(2*x**7 - 3*x**6 + 16*x**5 - 31*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)-2*x^7+3*x^6-16*x^5+31*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)-2 x^{7}+3 x^{6}-16 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-6+1/60*n^5-1/4*n^4+7/12*n^3-13/4*n^2-21/10*n+6*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -6+\frac{n^{5}}{60}-\frac{n^{4}}{4}+\frac{7 n^{3}}{12}-\frac{13 n^{2}}{4}-\frac{21 n}{10}+6 \,2^{n} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 180
a(7) = 468
a(n+1) = 2*a(n)-1/60*(n-1)*(n^4-19*n^3+66*n^2-154*n+60), 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 180
a \! \left(7\right) = 468
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-19 n^{3}+66 n^{2}-154 n +60\right)}{60}, \quad n \geq 8
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/21288/
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[29,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[5,x]+F[8,x]
F[8,x] = F[13,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]+F[8,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[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[13,x]*F[16,x]
F[16,x] = F[17,x]+F[24,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[13,x]*F[22,x]
F[22,x] = F[11,x]+F[19,x]
F[23,x] = F[13,x]*F[18,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[20,x]+F[26,x]+F[28,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[11,x]+F[25,x]
F[28,x] = F[13,x]*F[24,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[33,x]+F[40,x]
F[33,x] = F[11,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[11,x]*F[38,x]
F[38,x] = F[18,x]+F[39,x]
F[39,x] = F[10,x]+F[24,x]
F[40,x] = F[41,x]+F[58,x]
F[41,x] = F[42,x]+F[49,x]
F[42,x] = F[11,x]*F[43,x]
F[43,x] = F[1,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[13,x]*F[46,x]
F[46,x] = F[43,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[10,x]^2*F[13,x]
F[49,x] = F[19,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[49,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[11,x]^2*F[39,x]
F[56,x] = F[57,x]
F[57,x] = F[11,x]^3*F[10,x]
F[58,x] = F[25,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[13,x]*F[61,x]
F[61,x] = F[58,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[11,x]^2*F[64,x]
F[64,x] = F[17,x]+F[65,x]
F[65,x] = F[10,x]*F[11,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_{29}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{8}\! \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_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \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_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{58}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)
F_{49}\! \left(x \right) = F_{19}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \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_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{39}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11} \left(x \right)^{3} F_{10}\! \left(x \right)
F_{58}\! \left(x \right) = F_{25}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{17}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \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_29(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_5(x) + F_8(x))
Eq(F_8(x), F_13(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x) + F_8(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))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_13(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_24(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_13(x)*F_22(x))
Eq(F_22(x), F_11(x) + F_19(x))
Eq(F_23(x), F_13(x)*F_18(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_20(x) + F_26(x) + F_28(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_13(x)*F_24(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_40(x))
Eq(F_33(x), F_11(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_11(x)*F_38(x))
Eq(F_38(x), F_18(x) + F_39(x))
Eq(F_39(x), F_10(x) + F_24(x))
Eq(F_40(x), F_41(x) + F_58(x))
Eq(F_41(x), F_42(x) + F_49(x))
Eq(F_42(x), F_11(x)*F_43(x))
Eq(F_43(x), F_1(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_13(x)*F_46(x))
Eq(F_46(x), F_43(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_10(x)**2*F_13(x))
Eq(F_49(x), F_19(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_49(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_11(x)**2*F_39(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_10(x)*F_11(x)**3)
Eq(F_58(x), F_25(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_13(x)*F_61(x))
Eq(F_61(x), F_58(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_11(x)**2*F_64(x))
Eq(F_64(x), F_17(x) + F_65(x))
Eq(F_65(x), F_10(x)*F_11(x))
Pack JSON:
{"expansion_strats": [[{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_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": [1, 2, 3, 0], "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, 1, 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": [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": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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": [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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 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