0132_0213_1302_2301
Counting sequence:
1, 1, 2, 6, 20, 63, 183, 498, 1295, 3270, 8113, 19943, 48852, 119718, 294268, 726662, 1804342, 4506803, 11323649, 28613126, 72681841, 185501546, 475435707, 1222981531, 3155762150, 8164615418, 21170314518, 54994009498, 143073519720, 372683049415, 971754407083, 2535869289898, 6621894823827, 17300824157758, 45220448113253, 118235966136927, 309229752266872, 808920714726718, 2116432880293904, 5538110183431806, 14493224745593306, 37931943326615691, 99282814024965237, 259875816818064606, 680261073545530805, 1780735880004608898, 4661594722747422703, 12203327008609854323, 31946908559454426570, 83634372813753804210, 218950017432319371178, 573203013109252350578, 1500633126197580322252, 3928643448187867036223, 10285189131974963919263, 26926703271355283593442, 70494470322128149839127, 184555788960705182371958, 483171023062542411181209, 1264953461174438041023511, 3311681578240615615706396, 8670075420876720461985686, 22698512402587416778573092, 59425396070359767283495606, 155577542069597350677706110, 407306958048957197533780003, 1066342778674952030637131113, 2791720252724510398095012598, 7308815692102314023663556345, 19134722175002925098088701530, 50095341388173495531312175523, 131151282804903724837914201483, 343358468067014195307857475502, 898924042296500273019100957738, 2353413498262026206181508196798, 6161316126646291027519903955082, 16130534220545539274627873821232, 42230285193838245661384476892023, 110560318640886103576609913841939, 289450665213096013076699655115866, 757791665815838104218169185555611, 1983924309567059181803510875895406, 5193981216946158295836454803384397, 13598019248184127595379407081974031, 35600076339013796630419615188469440, 93202209386836703297656619276381358, 244006551047783788709187570828824248, 638817442189746800609346682790993198, 1672445772349235262450065523115259170, 4378519868436145032947939710515690875, 11463113819960829423897260721991126493, 30010821565140117403929511613851420238, 78569350822228101098619922298900918077, 205697230793843402474102171366627656178, 538522341341424659409459663418736320983, 1409869792789723921768681441025493221347, 3691087036136430277611110861730374044562, 9663391313817126213865137463912504189538, 25299086901670452888328142000704116960594, 66233869383826013337292705142000259467746, 173402521234912692570868277957710398941076
Generating function in Maple syntax:
-(9*x^7-46*x^6+99*x^5-125*x^4+95*x^3-42*x^2+10*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{9 x^{7}-46 x^{6}+99 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-9*x**7 + 46*x**6 - 99*x**5 + 125*x**4 - 95*x**3 + 42*x**2 - 10*x + 1)/((x - 1)**4*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^4*F(x)+9*x^7-46*x^6+99*x^5-125*x^4+95*x^3-42*x^2+10*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+9 x^{7}-46 x^{6}+99 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1 = 0
Explicit closed form in Maple syntax:
1/30*(-3*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(3*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)-1/6*n^3+2^(-1+n)*n+1/6*n-2^n+1
Explicit closed form in latex syntax:
\frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}-\frac{n^{3}}{6}+2^{-1+n} n +\frac{n}{6}-2^{n}+1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 183
a(7) = 498
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+1/6*n*(n^2-3*n-16), 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 183
a \! \left(7\right) = 498
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{n \left(n^{2}-3 n -16\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21195/
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[29,x]+F[8,x]
F[8,x] = F[23,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[19,x]+F[20,x]+F[22,x]
F[19,x] = 0
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[11,x]+F[18,x]
F[22,x] = F[13,x]*F[14,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[13,x]*F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[13,x]*F[17,x]*F[26,x]
F[29,x] = F[17,x]*F[30,x]
F[30,x] = F[31,x]+F[39,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[13,x]*F[34,x]
F[34,x] = F[10,x]+F[31,x]
F[35,x] = F[36,x]
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[15,x]+F[38,x]
F[38,x] = F[36,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[19,x]+F[41,x]+F[43,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[32,x]+F[40,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[14,x]+F[39,x]
F[45,x] = F[46,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[49,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = F[15,x]+F[48,x]
F[51,x] = F[46,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_{29}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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_{10}\! \left(x \right) F_{13}\! \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_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{39}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \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_{15}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \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_29(x) + F_8(x))
Eq(F_8(x), F_23(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_22(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_11(x) + F_18(x))
Eq(F_22(x), F_13(x)*F_14(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_13(x)*F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_13(x)*F_17(x)*F_26(x))
Eq(F_29(x), F_17(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_39(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_13(x)*F_34(x))
Eq(F_34(x), F_10(x) + F_31(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_15(x) + F_38(x))
Eq(F_38(x), F_36(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_19(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_32(x) + F_40(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_14(x) + F_39(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_15(x) + F_48(x))
Eq(F_51(x), F_46(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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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