0132_0213_1203_2031_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 139, 331, 741, 1580, 3246, 6492, 12755, 24806, 48041, 93053, 180765, 352684, 691396, 1361542, 2691755, 5338518, 10613863, 21140751, 42163813, 84170812, 168135546, 336003632, 671664051, 1342892214, 2685236069, 5369788289, 10738730621, 21476422556, 42951578640, 85901623066, 171801398827, 343600585990, 687198538211, 1374393955731, 2748784231333, 5497564142220, 10995123233734, 21990240586756, 43980474352467, 87960940821830, 175921872564513, 351843734706693, 703687457486621, 1407374901365708, 2814749787250652, 5629499556937646, 11258999094000747, 22517998165568566, 45035996305877599, 90071992583378839, 180143985134950949, 360287970234326556, 720575940428944786, 1441151880813656280, 2882303761578133171, 5764607523101688982, 11529215046142918365, 23058430092218976521, 46116860184364137981, 92233720368646914108, 184467440737204288040, 368934881474310184514, 737869762948512409451, 1475739525896906529062, 2951479051793683627995, 5902958103587225825563, 11805916207174297308133, 23611832414348426393836, 47223664828696669661918, 94447329657393140211372, 188894659314786064178131, 377789318629571893769382, 755578637259143533332121, 1511157274518286791490189, 3022314549036573285418269, 6044629098073146249389804, 12089258196146292151872756, 24178516392292583929721046, 48357032784585167456557355, 96714065569170334479538646, 193428131138340668492887127, 386856262276681336484952095, 773712524553362672432333541, 1547425049106725344288129276, 3094850098213450687958429098, 6189700196426901375255303040, 12379400392853802749802777779, 24758800785707605498848789366, 49517601571415210996889088597, 99035203142830421992915051665, 198070406285660843984909301373, 396140812571321687968836949468, 792281625142643375936628081216, 1584563250285286751872142724458, 3169126500570573503743100787627
Generating function in Maple syntax:
(3*x^8-11*x^7+24*x^6-47*x^5+63*x^4-54*x^3+28*x^2-8*x+1)/(2*x-1)/(x-1)^7
Generating function in latex syntax:
\frac{3 x^{8}-11 x^{7}+24 x^{6}-47 x^{5}+63 x^{4}-54 x^{3}+28 x^{2}-8 x +1}{\left(2 x -1\right) \left(x -1\right)^{7}}
Generating function in sympy syntax:
(3*x**8 - 11*x**7 + 24*x**6 - 47*x**5 + 63*x**4 - 54*x**3 + 28*x**2 - 8*x + 1)/((x - 1)**7*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^7*F(x)-3*x^8+11*x^7-24*x^6+47*x^5-63*x^4+54*x^3-28*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{7} F \! \left(x \right)-3 x^{8}+11 x^{7}-24 x^{6}+47 x^{5}-63 x^{4}+54 x^{3}-28 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-3+5*2^(-1+n)-7/144*n^4+7/16*n^3-883/360*n^2+16/15*n-1/240*n^5+1/720*n^6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -3+5 \,2^{-1+n}-\frac{7 n^{4}}{144}+\frac{7 n^{3}}{16}-\frac{883 n^{2}}{360}+\frac{16 n}{15}-\frac{n^{5}}{240}+\frac{n^{6}}{720} & \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) = 139
a(7) = 331
a(8) = 741
a(n+1) = 2*a(n)-1/720*(-1+n)*(n-10)*(n^4+2*n^3-23*n^2+192*n-144), 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) = 54
a \! \left(6\right) = 139
a \! \left(7\right) = 331
a \! \left(8\right) = 741
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-1+n \right) \left(n -10\right) \left(n^{4}+2 n^{3}-23 n^{2}+192 n -144\right)}{720}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18412/
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[31,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[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[14,x]
F[30,x] = F[26,x]
F[31,x] = F[32,x]+F[33,x]+F[52,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[46,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[13,x]+F[39,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[17,x]+F[43,x]
F[43,x] = F[40,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[39,x]
F[46,x] = F[31,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[36,x]
F[51,x] = F[47,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[66,x]
F[54,x] = F[21,x]+F[55,x]
F[55,x] = F[32,x]+F[56,x]+F[65,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[10,x]+F[18,x]
F[59,x] = F[55,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[18,x]
F[64,x] = F[60,x]
F[65,x] = F[12,x]*F[54,x]
F[66,x] = F[67,x]+F[72,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[21,x]
F[71,x] = F[67,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[66,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_{31}\! \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_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{46}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{31}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{36}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{66}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{32}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{72}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{21}\! \left(x \right)
F_{71}\! \left(x \right) = F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \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_31(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) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_14(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_52(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_46(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_13(x) + F_39(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_17(x) + F_43(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_39(x))
Eq(F_46(x), F_31(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_36(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_66(x))
Eq(F_54(x), F_21(x) + F_55(x))
Eq(F_55(x), F_32(x) + F_56(x) + F_65(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_10(x) + F_18(x))
Eq(F_59(x), F_55(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_18(x))
Eq(F_64(x), F_60(x))
Eq(F_65(x), F_12(x)*F_54(x))
Eq(F_66(x), F_67(x) + F_72(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_21(x))
Eq(F_71(x), F_67(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_66(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, 2, 0, 3], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 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