0213_1320_2130
Counting sequence:
1, 1, 2, 6, 21, 76, 270, 927, 3074, 9886, 30985, 95064, 286558, 851203, 2497550, 7252494, 20874861, 59630404, 169225518, 477513639, 1340705306, 3747697726, 10435070737, 28954040496, 80087091646, 220897122571, 607726482470, 1668084221742, 4568859998709, 12489795988636, 34082296120590, 92851499272239, 252574459546418, 686085856631518, 1861226005516825, 5043013310611464, 13648547368128286, 36899433738316819, 99659486778249278, 268911523525384014, 724963047811764861, 1952809427339404276, 5256073517114787822, 14136405814567926711, 37993523013904438346, 102044312395598525566, 273899495785074637345, 734734297016001812832, 1969783732596434714494, 5277977896270553910811, 14134752816479710378070, 37834728560677240789614, 101224174872238449461253, 270693575757438616703308, 723569152875292211033550, 1933295911348081379498559, 5163452079643466088492386, 13785178821547813770191134, 36789306422465147242134313, 98146288172364038189051064, 261742979452883808117267166, 697799366966887062534223651, 1859711851296010696262710382, 4954789661414627557124624526, 13196960830038850764766548621, 35139550452398907602911542436, 93539759714992899906804420270, 248930478660058008950838549447, 662285857035120981804014034554, 1761578884898326748645032452286, 4684371994470345774708781829425, 12453648896933870152799608898064, 33100988895669700620605292421438, 87960448591440267373519832129323, 233689335086949547418701881609350, 620723360500022226698629749684654, 1648419179710827341655180436400661, 4376733674723310772282064700681724, 11618441899490616086052915762745998, 30836372692865556640036107222825039, 81827358131652273776849520684691346, 217097966891066252547090860473381726, 575886656156831921856075134763919417, 1527370077238124227868934304023699816, 4050227688921966245761318871418802846, 10738517253969609801516683607410490931, 28466932752962450660918500298330867870, 75451902780431683615197212133151872462, 199956032234956865423760054490701727389, 529828342088913823818068316062886276436, 1403698181878817188034680078281722610030, 3718361618924624802651424125412732282839, 9848503733180214280041177768470235261098, 26081405340096259065530341455321154525438, 69061362507267842760720814093773707740225, 182845377082712295361322483321167620690624, 484037203223739697193618240422250420383486, 1281210841136143442246899518234577365601339, 3390866711344789334880696248849136743755958, 8973258857831002874612441148233194658456814, 23743247165786693204762309814056658174500901
Generating function in Maple syntax:
-(3*x^5-19*x^4+32*x^3-24*x^2+8*x-1)/(-1+x)/(-1+2*x)/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{3 x^{5}-19 x^{4}+32 x^{3}-24 x^{2}+8 x -1}{\left(-1+x \right) \left(-1+2 x \right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-3*x**5 + 19*x**4 - 32*x**3 + 24*x**2 - 8*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(-1+x)*(-1+2*x)*(x^2-3*x+1)^2*F(x)+3*x^5-19*x^4+32*x^3-24*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(-1+x \right) \left(-1+2 x \right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+3 x^{5}-19 x^{4}+32 x^{3}-24 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
1/50*(20*n-37*5^(1/2)-25)*(3/2-1/2*5^(1/2))^(-n)+1/50*(20*n+37*5^(1/2)-25)*(3/2+1/2*5^(1/2))^(-n)+3*2^n-1
Explicit closed form in latex syntax:
\frac{\left(20 n -37 \sqrt{5}-25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{\left(20 n +37 \sqrt{5}-25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}+3 \,2^{n}-1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)+1, n >= 6
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) = 21
a \! \left(5\right) = 76
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)+1, \quad n \geq 6
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/336/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[10,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[52,x]
F[4,x] = F[10,x]*F[5,x]
F[5,x] = 2*F[4,x]+F[1,x]+F[6,x]
F[6,x] = F[10,x]*F[7,x]
F[7,x] = F[1,x]+F[24,x]+F[26,x]+F[50,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[11,x]*F[14,x]
F[10,x] = x
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[11,x]
F[14,x] = F[1,x]+F[15,x]+F[16,x]
F[15,x] = F[10,x]*F[14,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[11,x]*F[18,x]*F[21,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[11,x]*F[18,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[10,x]*F[14,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = 2*F[24,x]+F[1,x]+F[26,x]+F[27,x]+F[8,x]
F[26,x] = F[10,x]*F[7,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[11,x]*F[29,x]
F[29,x] = F[30,x]+F[48,x]
F[30,x] = F[1,x]+F[31,x]+F[32,x]+F[34,x]
F[31,x] = F[10,x]*F[30,x]
F[32,x] = F[33,x]
F[33,x] = F[10,x]*F[11,x]*F[18,x]*F[21,x]
F[34,x] = F[10,x]*F[35,x]
F[35,x] = F[1,x]+F[34,x]+F[36,x]+F[38,x]
F[36,x] = F[37,x]
F[37,x] = F[11,x]^2*F[10,x]
F[38,x] = F[39,x]
F[39,x] = F[10,x]*F[14,x]*F[40,x]
F[40,x] = F[11,x]+F[41,x]
F[41,x] = F[12,x]+F[42,x]
F[42,x] = F[43,x]+F[44,x]+F[47,x]
F[43,x] = 0
F[44,x] = F[10,x]*F[45,x]
F[45,x] = F[12,x]+F[46,x]
F[46,x] = F[44,x]
F[47,x] = F[10,x]*F[41,x]
F[48,x] = F[49,x]
F[49,x] = F[10,x]*F[11,x]*F[18,x]*F[21,x]
F[50,x] = F[51,x]
F[51,x] = F[10,x]*F[11,x]*F[30,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[1,x]+F[50,x]+F[54,x]+F[56,x]
F[54,x] = F[10,x]*F[55,x]
F[55,x] = 2*F[54,x]+F[1,x]+F[27,x]+F[56,x]
F[56,x] = F[10,x]*F[53,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_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{52}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = 2 F_{4}\! \left(x \right)+F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{10}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)+F_{50}\! \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_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = x
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_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{8}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{48}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \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_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = 0
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right) F_{41}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{30}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{1}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{54}\! \left(x \right)+F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \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_10(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_52(x))
Eq(F_4(x), F_10(x)*F_5(x))
Eq(F_5(x), F_1(x) + 2*F_4(x) + F_6(x))
Eq(F_6(x), F_10(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_24(x) + F_26(x) + F_50(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_11(x)*F_14(x))
Eq(F_10(x), 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_11(x))
Eq(F_14(x), F_1(x) + F_15(x) + F_16(x))
Eq(F_15(x), F_10(x)*F_14(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_11(x)*F_18(x)*F_21(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_11(x)*F_18(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_10(x)*F_14(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_1(x) + 2*F_24(x) + F_26(x) + F_27(x) + F_8(x))
Eq(F_26(x), F_10(x)*F_7(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_11(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_48(x))
Eq(F_30(x), F_1(x) + F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), F_10(x)*F_30(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_10(x)*F_11(x)*F_18(x)*F_21(x))
Eq(F_34(x), F_10(x)*F_35(x))
Eq(F_35(x), F_1(x) + F_34(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_10(x)*F_11(x)**2)
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_10(x)*F_14(x)*F_40(x))
Eq(F_40(x), F_11(x) + F_41(x))
Eq(F_41(x), F_12(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_44(x) + F_47(x))
Eq(F_43(x), 0)
Eq(F_44(x), F_10(x)*F_45(x))
Eq(F_45(x), F_12(x) + F_46(x))
Eq(F_46(x), F_44(x))
Eq(F_47(x), F_10(x)*F_41(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_10(x)*F_11(x)*F_18(x)*F_21(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_10(x)*F_11(x)*F_30(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_1(x) + F_50(x) + F_54(x) + F_56(x))
Eq(F_54(x), F_10(x)*F_55(x))
Eq(F_55(x), F_1(x) + F_27(x) + 2*F_54(x) + F_56(x))
Eq(F_56(x), F_10(x)*F_53(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, 2, 1, 3], "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, 1, 3, 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], "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": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, 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