0132_1032_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 182, 518, 1449, 4012, 11036, 30220, 82474, 224490, 609743, 1653168, 4475264, 12098642, 32669102, 88119758, 237458613, 639317524, 1719844292, 4623073816, 12418340050, 33335452242, 89428522907, 239766032088, 642472619624, 1720636661342, 4605772044566, 12322665123830, 32953778492289, 88087215979900, 235361682727148, 628610189542948, 1678251070904314, 4478875073352762, 11948783819732327, 31865964757461888, 84954148717058768, 226413073455574634, 603229740851363582, 1606693427202797726, 4278157430990269005, 11388281708607631972, 30306888233606627732, 80632479566671375408, 214470635252964359074, 570319575181340137122, 1516228434879668453171, 4030046579050011900840, 10709213436090050706104, 28451819140349355553910, 75573617803051001797478, 200696929749003438606278, 532873482940565787640089, 1414564555830513742717132, 3754386978824228794731452, 9962655717696913630993468, 26432191373143083203095498, 70115692625278511284352394, 185961597902396811425365247, 493127460913363641936038736, 1307449152644114534288521760, 3465946740030328749399120002, 9186542927521576154805000782, 24345410877441321240021082478, 64508724344836761680950296549, 170906137233041792588045980852, 452726777923725984249434565476, 1199100093133581947726363197000, 3175534100620132877166886343794, 8408558109413135998330207894770, 22262347330239972138074889377483, 58934049287892256351733298636408, 155994289649391936083672958387464, 412856721599202310221655771087502, 1092545091844293763898987341490102, 2890878302073551798140275728238998, 7648419842081011845267576700325553, 20233271559105782633371738307467804, 53519735812265024663503838845750412, 141552068473687942559947267519232596, 374346526419463836561200573662945690, 989891548620095553645627479445797082, 2617330176135124824577123703477318807, 6919701112030375082528866863069174048, 18292577499992151206858946226477549424, 48352842275799763320596339573124848282, 127799577650912699505611872355638472606, 337751964759562647039508641191663029310, 892540910552065245139585006183045322109, 2358418474586724044611079206016322206980, 6231273042353780268852658797810049620788, 16462528532220926976172440226449912789088, 43489137664401019982142999432428919701314, 114876231911509404843387456923918654797122, 303420775311612039084776588826583642685027, 801358398297244070627768780122321065766728, 2116290115160367595871623379668190948158808
Generating function in Maple syntax:
-(x^8-7*x^7+15*x^6-26*x^5+50*x^4-56*x^3+32*x^2-9*x+1)/(2*x-1)/(x-1)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{x^{8}-7 x^{7}+15 x^{6}-26 x^{5}+50 x^{4}-56 x^{3}+32 x^{2}-9 x +1}{\left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-x**8 + 7*x**7 - 15*x**6 + 26*x**5 - 50*x**4 + 56*x**3 - 32*x**2 + 9*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^2*(x^2-3*x+1)^2*F(x)+x^8-7*x^7+15*x^6-26*x^5+50*x^4-56*x^3+32*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{8}-7 x^{7}+15 x^{6}-26 x^{5}+50 x^{4}-56 x^{3}+32 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/100*((-30*n-2)*5^(1/2)+70*n+50)*(3/2-1/2*5^(1/2))^(-n)+1/100*((30*n+2)*5^(1/2)+70*n+50)*(3/2+1/2*5^(1/2))^(-n)-n-1/4*2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\left(-30 n -2\right) \sqrt{5}+70 n +50\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{100}+\\\frac{\left(\left(30 n +2\right) \sqrt{5}+70 n +50\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{100}-n -\frac{2^{n}}{4}+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) = 20
a(5) = 62
a(6) = 182
a(7) = 518
a(8) = 1449
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)+n, 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 182
a \! \left(7\right) = 518
a \! \left(8\right) = 1449
a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)+n, \quad n \geq 9
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21277/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[34,x]+F[4,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[1,x]+F[19,x]+F[33,x]+F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[1,x]+F[19,x]+F[31,x]+F[6,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[14,x]*F[16,x]
F[10,x] = F[11,x]+F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = x
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]+F[29,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[14,x]*F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[14,x]*F[27,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[1,x]+F[25,x]+F[29,x]
F[29,x] = F[14,x]*F[28,x]
F[30,x] = F[9,x]
F[31,x] = F[14,x]*F[32,x]
F[32,x] = F[1,x]+F[18,x]+F[19,x]+F[31,x]+F[6,x]
F[33,x] = F[14,x]*F[5,x]
F[34,x] = F[14,x]*F[35,x]
F[35,x] = F[1,x]+F[36,x]+F[37,x]+F[49,x]
F[36,x] = F[14,x]*F[35,x]
F[37,x] = F[38,x]
F[38,x] = F[14,x]*F[39,x]
F[39,x] = F[28,x]+F[40,x]+F[48,x]
F[40,x] = F[14,x]*F[16,x]*F[41,x]
F[41,x] = F[22,x]+F[42,x]
F[42,x] = F[17,x]+F[43,x]
F[43,x] = F[44,x]+F[45,x]+F[47,x]
F[44,x] = 0
F[45,x] = F[14,x]*F[46,x]
F[46,x] = F[23,x]+F[43,x]
F[47,x] = F[14,x]*F[42,x]
F[48,x] = F[14,x]*F[16,x]*F[35,x]
F[49,x] = F[14,x]*F[35,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{6}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right)
F_{30}\! \left(x \right) = F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{28}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \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_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = 0
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right) F_{42}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{35}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_34(x) + F_4(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_19(x) + F_33(x) + F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_19(x) + F_31(x) + F_6(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_14(x)*F_16(x))
Eq(F_10(x), F_11(x) + F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_14(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x) + F_29(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_14(x)*F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_14(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_1(x) + F_25(x) + F_29(x))
Eq(F_29(x), F_14(x)*F_28(x))
Eq(F_30(x), F_9(x))
Eq(F_31(x), F_14(x)*F_32(x))
Eq(F_32(x), F_1(x) + F_18(x) + F_19(x) + F_31(x) + F_6(x))
Eq(F_33(x), F_14(x)*F_5(x))
Eq(F_34(x), F_14(x)*F_35(x))
Eq(F_35(x), F_1(x) + F_36(x) + F_37(x) + F_49(x))
Eq(F_36(x), F_14(x)*F_35(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_14(x)*F_39(x))
Eq(F_39(x), F_28(x) + F_40(x) + F_48(x))
Eq(F_40(x), F_14(x)*F_16(x)*F_41(x))
Eq(F_41(x), F_22(x) + F_42(x))
Eq(F_42(x), F_17(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_45(x) + F_47(x))
Eq(F_44(x), 0)
Eq(F_45(x), F_14(x)*F_46(x))
Eq(F_46(x), F_23(x) + F_43(x))
Eq(F_47(x), F_14(x)*F_42(x))
Eq(F_48(x), F_14(x)*F_16(x)*F_35(x))
Eq(F_49(x), F_14(x)*F_35(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, 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