0213_0231_1320_3012_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 141, 371, 973, 2549, 6675, 17477, 45757, 119795, 313629, 821093, 2149651, 5627861, 14733933, 38573939, 100987885, 264389717, 692181267, 1812154085, 4744280989, 12420688883, 32517785661, 85132668101, 222880218643, 583507987829, 1527643744845, 3999423246707, 10470625995277, 27412454739125, 71766738222099, 187887759927173, 491896541559421, 1287801864751091, 3371509052693853, 8826725293330469, 23108666827297555, 60499275188562197, 158389158738389037, 414668201026604915, 1085615444341425709, 2842178131997672213, 7440918951651590931, 19480578722957100581, 51000817217219710813, 133521872928702031859, 349564801568886384765, 915172531777957122437, 2395952793764984982547, 6272685849516997825205, 16422104754786008493069, 42993628414841027654003, 112558780489737074468941, 294682713054370195752821, 771489358673373512789523, 2019785362965750342615749, 5287866730223877515057725, 13843814827705882202557427, 36243577752893769092614557, 94886918430975425075286245, 248417177540032506133244179, 650364614189122093324446293, 1702676665027333773840094701, 4457665380892879228195837811, 11670319477651303910747418733, 30553293052061032504046418389, 79989559678531793601391836435, 209415385983534348300129090917, 548256598272071251298995436317, 1435354408832679405596857218035, 3757806628225966965491576217789, 9838065475845221490877871435333, 25756389799309697507142038088211, 67431103922083871030548242829301, 176536921966941915584502690399693, 462179661978741875722959828369779, 1210002063969283711584376794709645, 3167826529929109259030170555759157, 8293477525818044065506134872567827, 21712606047525022937488234061944325, 56844340616757024746958567313265149, 148820415802746051303387467877851123, 389616906791481129163203836320288221, 1020030304571697336186224041083013541, 2670474006923610879395468286928752403, 6991391716199135302000180819703243669, 18303701141673795026605074172180978605, 47919711708822249777815041696839692147, 125455433984792954306840050918338097837, 328446590245556613142705111058174601365, 859884336751876885121275282256185706259, 2251206420010074042221120735710382517413, 5893734923278345241542086924874961845981, 15429998349824961682405140038914503020531, 40396260126196539805673333191868547215613, 105758782028764657734614859536691138626309, 276880085960097433398171245418204868663315
Generating function in Maple syntax:
(x^6+x^5-2*x^4-x^3-2*x^2+3*x-1)/(x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{6}+x^{5}-2 x^{4}-x^{3}-2 x^{2}+3 x -1}{\left(x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**6 + x**5 - 2*x**4 - x**3 - 2*x**2 + 3*x - 1)/((x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2-3*x+1)*F(x)+x^6+x^5-2*x^4-x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{6}+x^{5}-2 x^{4}-x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,36,n = 1,14,n = 2,5,n = 3,1,0)-119/5*(3/2-1/2*5^(1/2))^(1-n)-119/5*(3/2+1/2*5^(1/2))^(1-n)+1/10*(-41*5^(1/2)+187)*(3/2-1/2*5^(1/2))^(-n)-1+1/10*(41*5^(1/2)+187)*(3/2+1/2*5^(1/2))^(-n)
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}36 & n =0 \\ 14 & n =1 \\ 5 & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-\frac{119 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}}{5}-\frac{119 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}}{5}+\frac{\left(-41 \sqrt{5}+187\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}-1+\frac{\left(41 \sqrt{5}+187\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 141
a(n+2) = -a(n)+3*a(n+1)+1, 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) = 53
a \! \left(6\right) = 141
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20434/
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[5,x]
F[4,x] = x
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[11,x]
F[15,x] = F[16,x]+F[27,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[13,x]+F[19,x]
F[19,x] = F[16,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[26,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[4,x]
F[25,x] = F[20,x]
F[26,x] = F[16,x]*F[4,x]
F[27,x] = F[21,x]+F[28,x]+F[50,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[38,x]
F[30,x] = F[31,x]+F[7,x]
F[31,x] = F[21,x]+F[32,x]+F[33,x]
F[32,x] = F[4,x]*F[7,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[4,x]
F[36,x] = F[37,x]
F[37,x] = x^2
F[38,x] = F[27,x]+F[39,x]
F[39,x] = F[21,x]+F[40,x]+F[44,x]+F[45,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[31,x]
F[43,x] = F[39,x]
F[44,x] = F[27,x]*F[4,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[24,x]+F[49,x]
F[49,x] = F[47,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[15,x]+F[52,x]
F[52,x] = F[47,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{11}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)+F_{50}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = x^{2}
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{31}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{24}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{47}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_11(x))
Eq(F_15(x), F_16(x) + F_27(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_13(x) + F_19(x))
Eq(F_19(x), F_16(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_26(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_4(x))
Eq(F_25(x), F_20(x))
Eq(F_26(x), F_16(x)*F_4(x))
Eq(F_27(x), F_21(x) + F_28(x) + F_50(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_38(x))
Eq(F_30(x), F_31(x) + F_7(x))
Eq(F_31(x), F_21(x) + F_32(x) + F_33(x))
Eq(F_32(x), F_4(x)*F_7(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), x**2)
Eq(F_38(x), F_27(x) + F_39(x))
Eq(F_39(x), F_21(x) + F_40(x) + F_44(x) + F_45(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_31(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_27(x)*F_4(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_24(x) + F_49(x))
Eq(F_49(x), F_47(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_15(x) + F_52(x))
Eq(F_52(x), F_47(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, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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