0132_0231_0321_1302_2031_3120
Counting sequence:
1, 1, 2, 6, 18, 50, 132, 336, 830, 1998, 4704, 10868, 24714, 55458, 123068, 270552, 590070, 1278230, 2752824, 5898588, 12583298, 26739114, 56623572, 119538176, 251658798, 528482910, 1107296912, 2315256516, 4831838970, 10066330418, 20937966444, 43486544808, 90194314214, 186831078438, 386547057768, 798863918252, 1649267442930, 3401614099770, 7009386628484, 14431090116048, 29686813951518, 61022895343214, 125344325568192, 257285720901396, 527765581334378, 1081919441733570, 2216615441598492, 4538783999461496, 9288674231453910, 18999560927971638, 38843546786072984, 79375943432407548, 162129586585340514, 331014572611734218, 675539944105577268, 1378101485975374752, 2810246167479192590, 5728578726015274110, 11673330234144328944, 23779006032516222308, 48422703193487576538, 98574788643885420114, 200608341801591377612, 408134212630823833416, 830103483316929826758, 1687877082744423977030, 3431094397709976604872, 6972869259862210515276, 14167099448608935645650, 28776920754986900525658, 58439285225511859524324, 118649457882099835999088, 240840690626351905903614, 488764930977008279622798, 991696961402625494881568, 2011728121702468861040052, 4080124641199373464639050, 8273586077987618414401250, 16773845747152979799054204, 34001038676661445538617368, 68908771718033862958258358, 139630932165489669678569814, 282888641789823226881251832, 573030838497334228810734236, 1160568786830044007717935938, 2350151793330839115628813290, 4758332026003180431643516052, 9632720930689365264058817856, 19497555618744739329661214190, 39459338752221496262409592478, 79847132533907027730993520464, 161551175126742125874335719428, 326816170371340392573368803514, 661059980978393066796132344178, 1336975242428210696891054170668, 2703661045799270520379687314152, 5466743213484239293954532582310, 11052328670739875094299381081190, 22342341829022543201379394004264, 45160052633130672428320051701228, 91270843216432516907762630796978
Generating function in Maple syntax:
-(2*x^6+5*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{6}+5 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:
(-2*x**6 - 5*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)+2*x^6+5*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)+2 x^{6}+5 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,6+3/4*2^n*n-3*2^n-n+n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 6+\frac{3 \,2^{n} n}{4}-3 \,2^{n}-n +n^{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 50
a(6) = 132
a(n+2) = n^2-4*a(n)+4*a(n+1)-5*n+8, n >= 7
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) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 132
a \! \left(n +2\right) = n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-5 n +8, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22340/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[38,x]+F[8,x]
F[8,x] = F[33,x]+F[9,x]
F[9,x] = F[10,x]*F[24,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[15,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[15,x]*F[20,x]*F[21,x]
F[20,x] = F[1,x]+F[13,x]
F[21,x] = F[15,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[15,x]*F[21,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[15,x]*F[26,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[13,x]*F[26,x]
F[28,x] = F[29,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[28,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[15,x]*F[24,x]
F[33,x] = F[16,x]*F[20,x]*F[34,x]
F[34,x] = F[1,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[31,x]+F[34,x]
F[38,x] = F[34,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[13,x]*F[16,x]*F[20,x]*F[21,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_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \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_{38}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{33}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{24}\! \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_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
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_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{20}\! \left(x \right) F_{21}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{21}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_38(x) + F_8(x))
Eq(F_8(x), F_33(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_24(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_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_15(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_15(x)*F_20(x)*F_21(x))
Eq(F_20(x), F_1(x) + F_13(x))
Eq(F_21(x), F_15(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_15(x)*F_21(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_15(x)*F_26(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_13(x)*F_26(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_28(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_15(x)*F_24(x))
Eq(F_33(x), F_16(x)*F_20(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_31(x) + F_34(x))
Eq(F_38(x), F_34(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_13(x)*F_16(x)*F_20(x)*F_21(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, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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": [[{"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": [0, 2, 3, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 1]]}, 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