0213_0231_0321_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 58, 167, 456, 1193, 3018, 7435, 17932, 42509, 99342, 229391, 524304, 1187857, 2670610, 5963795, 13238292, 29229077, 64225302, 140509207, 306184216, 664797209, 1438646298, 3103784987, 6677331996, 14327742493, 30668750878, 65498251295, 139586437152, 296889614369, 630286450722, 1335734829091, 2826088480804, 5970004541477, 12592844111910, 26525718020135, 55800215109672, 117235427311657, 246015726714922, 515670953426987, 1079720418476076, 2258396883451949, 4719103906414638, 9851624184873007, 20547673299877936, 42819380832108593, 89157198873100338, 185492009652322355, 385620718093598772, 801077783718527029, 1662954162406555702, 3449757314565799991, 7151716208264347704, 14816842774048931897, 30678520661647818810, 63482740347414511675, 131288936337104699452, 271368899146836607037, 560608081615079342142, 1157533190625274363967, 2388853357545386934336, 4927586510689663975489, 10159544298595535552578, 20937054523660341084227, 43128487644332931678276, 88802625970837781479493, 182770340282314237411398, 376018431198495500140615, 773287511569904403742792, 1589666617295994320060489, 3266697014525077076574282, 6710482772157765848662091, 13779865397013624733565004, 28286975232389174830039117, 58047328807433678966751310, 119079193232040973708558415, 244203015562155093290647632, 500646405047908306975195217, 1026075789397916512031866962, 2102322000309840134814040147, 4306193769467309120303398996, 8817904928269105200306847829, 18051680338485642836712620118, 36944773047423067579020738647, 75591713648863533036027773016, 154626448031989529961618735193, 316216308784959323969545044058, 646514185516789848566067626075, 1321500991937143443454815109212, 2700566195701057069692439494749, 5517498755094939885225396666462, 11270206117654102022681626935391, 23015781210393790070924517572704, 46992203891273035235170755543137, 95925497764145546741383337869410, 195772789572747178193647101280355, 399468395396920790146648597594212, 814940879621722976487193073156197
Generating function in Maple syntax:
(2*x^6-11*x^4+23*x^3-19*x^2+7*x-1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
\frac{2 x^{6}-11 x^{4}+23 x^{3}-19 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(2*x**6 - 11*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**2*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(x-1)^2*(2*x-1)^3*F(x)+2*x^6-11*x^4+23*x^3-19*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)+2 x^{6}-11 x^{4}+23 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/16*(n^2+3*n-14)*2^n+n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n^{2}+3 n -14\right) 2^{n}}{16}+n +1 & \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) = 58
a(6) = 167
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-n+2, 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) = 19
a \! \left(5\right) = 58
a \! \left(6\right) = 167
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-n +2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21835/
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[29,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[5,x]+F[9,x]
F[9,x] = F[10,x]*F[26,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[14,x]+F[16,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[12,x]*F[14,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[25,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[15,x]
F[25,x] = F[12,x]*F[16,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[14,x]*F[26,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[63,x]
F[32,x] = F[33,x]+F[4,x]
F[33,x] = F[10,x]*F[34,x]
F[34,x] = F[26,x]+F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[27,x]
F[37,x] = F[21,x]+F[38,x]+F[53,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[24,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[40,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[21,x]+F[45,x]+F[47,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[24,x]+F[44,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[17,x]+F[44,x]
F[49,x] = 2*F[21,x]+F[50,x]+F[52,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[41,x]+F[49,x]
F[52,x] = F[12,x]*F[43,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[48,x]+F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[17,x]+F[56,x]
F[59,x] = 2*F[21,x]+F[60,x]+F[62,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[44,x]+F[59,x]
F[62,x] = F[12,x]*F[55,x]
F[63,x] = F[36,x]*F[64,x]
F[64,x] = F[10,x]+F[65,x]
F[65,x] = F[1,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_{12}\! \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_{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_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{63}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{26}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)+F_{53}\! \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_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{41}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{17}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{44}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{63}\! \left(x \right) = F_{36}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{1}\! \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_12(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_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_26(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_16(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_12(x)*F_14(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_25(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_15(x))
Eq(F_25(x), F_12(x)*F_16(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_14(x)*F_26(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_63(x))
Eq(F_32(x), F_33(x) + F_4(x))
Eq(F_33(x), F_10(x)*F_34(x))
Eq(F_34(x), F_26(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_27(x))
Eq(F_37(x), F_21(x) + F_38(x) + F_53(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_24(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_40(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_21(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_24(x) + F_44(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_17(x) + F_44(x))
Eq(F_49(x), 2*F_21(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_41(x) + F_49(x))
Eq(F_52(x), F_12(x)*F_43(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_48(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_17(x) + F_56(x))
Eq(F_59(x), 2*F_21(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_44(x) + F_59(x))
Eq(F_62(x), F_12(x)*F_55(x))
Eq(F_63(x), F_36(x)*F_64(x))
Eq(F_64(x), F_10(x) + F_65(x))
Eq(F_65(x), F_1(x) + F_11(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, 2, 1, 3], "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": [3, 0, 2, 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, 2, 1, 3], "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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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