0132_0213_0321_1032_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 331, 765, 1728, 3844, 8457, 18447, 39958, 86046, 184359, 393265, 835644, 1769544, 3735637, 7864419, 16515186, 34603138, 72351891, 150995109, 314572984, 654311628, 1358954721, 2818572535, 5838471438, 12079595814, 24964497727, 51539607897, 106300440948, 219043332496, 450971566509, 927712936395, 1906965479914, 3917010174474, 8040178778667, 16492674417229, 33809982554736, 69269232550548, 141836999983801, 290271069733599, 593736278999814, 1213860837065518, 2480498232263511, 5066549580792705, 10344205394117548, 21110623253300184, 43065671436731397, 87820192733725747, 179018085187978338, 364791569817011346, 743093938516133059, 1513209474796487925, 3080462145121420584, 6269010681299731804, 12754194144713246097, 25940733853654058439, 52746158835763250686, 107221699928436770358, 217902164370694080111, 442721857769029240489, 899278773593340643044, 1826227663297245611808, 3707795558815619876701, 7526271582073497061275, 15273904093031508740058, 30990530043832046716954, 62866503803202151909467, 127503895037480420771997, 258549564937113075452128, 524182679598530618722596, 1062532458645670173084009, 2153399116188558217447855, 4363466630171552177457654, 8840270055931975840041534, 17907213703041694650337927, 36267774588438875241188049, 73442243541588722363403036, 148697875812599388488862568, 301022529084042664501840821, 609298613085773104051915779, 1233104336006921758200302674, 2495222891684594616593550498, 5048474222710691433572994291, 10213005324104387267917778245, 20658124405574783337379138968, 41780476325881584277845446124, 84489407681227203761865231937, 170835725421382477936079146647, 345385270960621096696855662318, 698198182156954475043106066246, 1411251644785333513385001619359, 2852213850513516153367582216185, 5763848822912730559930322391124, 11646539889596857626250960703664, 23530764266736508265282553254157, 47536897508558602556126370206059
Generating function in Maple syntax:
-(4*x^7-x^6-3*x^5+6*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{7}-x^{6}-3 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**7 + x**6 + 3*x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+4*x^7-x^6-3*x^5+6*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{7}-x^{6}-3 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,9-11/2*n+1/2*n^2+3/8*2^n*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 9-\frac{11 n}{2}+\frac{n^{2}}{2}+\frac{3 \,2^{n} n}{8} & \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) = 331
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-15/2*n+20, 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) = 331
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{15 n}{2}+20, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21599/
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[8,x]
F[4,x] = F[38,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[20,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[8,x]
F[17,x] = F[18,x]+F[8,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[8,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[25,x]+F[26,x]+F[31,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[15,x]+F[28,x]
F[28,x] = F[25,x]+F[26,x]+F[29,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[21,x]+F[28,x]
F[31,x] = F[32,x]*F[8,x]
F[32,x] = F[30,x]+F[33,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[35,x]
F[35,x] = F[21,x]*F[8,x]
F[36,x] = F[37,x]
F[37,x] = F[33,x]*F[8,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[8,x]
F[40,x] = F[41,x]+F[55,x]
F[41,x] = F[4,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]*F[8,x]
F[44,x] = F[45,x]+F[52,x]
F[45,x] = F[46,x]+F[50,x]
F[46,x] = F[14,x]*F[47,x]
F[47,x] = F[1,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[23,x]*F[47,x]*F[8,x]
F[50,x] = F[51,x]
F[51,x] = F[8,x]^2*F[15,x]*F[23,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[14,x]*F[48,x]
F[54,x] = F[15,x]*F[8,x]
F[55,x] = F[56,x]
F[56,x] = F[57,x]*F[59,x]
F[57,x] = F[45,x]+F[58,x]
F[58,x] = F[48,x]+F[54,x]
F[59,x] = F[60,x]
F[60,x] = F[14,x]*F[61,x]*F[8,x]
F[61,x] = F[1,x]+F[62,x]
F[62,x] = F[59,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{38}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{55}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{1}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right) F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{8} \left(x \right)^{2} F_{15}\! \left(x \right) F_{23}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right) F_{48}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right) F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{45}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right) F_{61}\! \left(x \right) F_{8}\! \left(x \right)
F_{61}\! \left(x \right) = F_{1}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{59}\! \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_8(x))
Eq(F_4(x), F_38(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_20(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_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_8(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_8(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_31(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_15(x) + F_28(x))
Eq(F_28(x), F_25(x) + F_26(x) + F_29(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_21(x) + F_28(x))
Eq(F_31(x), F_32(x)*F_8(x))
Eq(F_32(x), F_30(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_21(x)*F_8(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_33(x)*F_8(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_8(x))
Eq(F_40(x), F_41(x) + F_55(x))
Eq(F_41(x), F_4(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x)*F_8(x))
Eq(F_44(x), F_45(x) + F_52(x))
Eq(F_45(x), F_46(x) + F_50(x))
Eq(F_46(x), F_14(x)*F_47(x))
Eq(F_47(x), F_1(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_23(x)*F_47(x)*F_8(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_15(x)*F_23(x)*F_8(x)**2)
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_14(x)*F_48(x))
Eq(F_54(x), F_15(x)*F_8(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_57(x)*F_59(x))
Eq(F_57(x), F_45(x) + F_58(x))
Eq(F_58(x), F_48(x) + F_54(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_14(x)*F_61(x)*F_8(x))
Eq(F_61(x), F_1(x) + F_62(x))
Eq(F_62(x), F_59(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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