0132_1203_3021
Counting sequence:
1, 1, 2, 6, 21, 73, 244, 787, 2468, 7570, 22809, 67727, 198664, 576775, 1659914, 4741254, 13454541, 37964341, 106590076, 297951451, 829614896, 2301930586, 6367200417, 17562228251, 48317182096, 132621950863, 363253065494, 993026186382, 2709804698133, 7382470135105, 20081940292804, 54550406490787, 147986380271804, 400973518724770, 1085212501752489, 2933936327166119, 7924139470763416, 21381847307477911, 57643972307583266, 155274448320395286, 417930007632536541, 1124043238247846221, 3021032338021870732, 8114024554635553771, 21779122130938913096, 58422615673549020778, 156628469988804043569, 419682764550875979827, 1123940006930952655648, 3008477870969727485215, 8048995337263081712174, 21524672860683816245022, 57536015954571387602469, 153730188975465063128665, 410584001305244837745364, 1096163354219897655111859, 2925401233363511334012116, 7804384331625397015926706, 20813288564782297023834681, 55487747822804382335739071, 147881217552667491551728936, 393997426466340316820181031, 1049404364378973866994379322, 2794254053213761036222342566, 7438179653512149594372087021, 19794712097834872062146542693, 52664416357890865203553362844, 140079488948244107104573162747, 372498446704608952240568705888, 990308087883372342599700182650, 2632178130954896334670669151169, 6994591010439265798355828383307, 18582936703022508117294594852784, 49359879801739426397636477276335, 131082343010050259171760032729414, 348038409451224830803405509676974, 923901025708932550682466612057141, 2452117828558129026856002075423601, 6506943802266704295209031326100196, 17263734444300955880992516451316931, 45794830786587387512835214284702956, 121457450817407022857943619271747266, 322077029115822136952124083174228937, 853935465979283402381190770900407511, 2263715349720720583268220841006413496, 6000006696432149723910256704257036983, 15900707098429685908124590033398714194, 42132525562179179989519767762100601142, 111623700119240055521055269656548857853, 295688655425654004592679124185912380285, 783165677517006182105916131699822933164, 2074028530573258937757748946865456352267, 5491856963187479080426257053022336053624, 14540133352496028225284764258764471227530, 38491379025837684851499620899108945079889, 101883920526123890055743682680062845194339, 269647297024322448999405736210842763095616, 713568797161111887787281151962453673681471, 1888104659830049850253814854721118160110686, 4995371051827918753884744323493509055522366, 13214840538779392276403297978930227394860101
Generating function in Maple syntax:
-(7*x^6-25*x^5+51*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{7 x^{6}-25 x^{5}+51 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:
(-7*x**6 + 25*x**5 - 51*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)+7*x^6-25*x^5+51*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)+7 x^{6}-25 x^{5}+51 x^{4}-56 x^{3}+32 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
1/50*((25*n+17)*5^(1/2)-45*n-75)*(3/2-1/2*5^(1/2))^(-n)+1/50*((-25*n-17)*5^(1/2)-45*n-75)*(3/2+1/2*5^(1/2))^(-n)-n+2^n+3
Explicit closed form in latex syntax:
\frac{\left(\left(25 n +17\right) \sqrt{5}-45 n -75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{\left(\left(-25 n -17\right) \sqrt{5}-45 n -75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-n +2^{n}+3
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 244
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)+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) = 21
a \! \left(5\right) = 73
a \! \left(6\right) = 244
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 -2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14719/
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[21,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[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[52,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[32,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]+F[6,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[29,x]
F[32,x] = F[22,x]+F[33,x]
F[33,x] = 2*F[15,x]+F[34,x]+F[38,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[26,x]
F[37,x] = F[33,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[6,x]
F[45,x] = F[41,x]
F[46,x] = 2*F[15,x]+F[47,x]+F[51,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[30,x]
F[50,x] = F[46,x]
F[51,x] = F[12,x]*F[40,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[66,x]
F[54,x] = F[2,x]+F[55,x]
F[55,x] = F[15,x]+F[56,x]+F[65,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[10,x]+F[30,x]
F[59,x] = F[55,x]+F[60,x]
F[60,x] = 2*F[15,x]+F[51,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[30,x]
F[64,x] = F[60,x]
F[65,x] = F[12,x]*F[54,x]
F[66,x] = F[41,x]+F[67,x]
F[67,x] = 2*F[15,x]+F[47,x]+F[68,x]
F[68,x] = F[12,x]*F[66,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_{21}\! \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_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{52}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{6}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \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_{26}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \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_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{30}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{66}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{30}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = F_{41}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \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_21(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_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_52(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_32(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x) + F_6(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_29(x))
Eq(F_32(x), F_22(x) + F_33(x))
Eq(F_33(x), 2*F_15(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_26(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_6(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), 2*F_15(x) + F_47(x) + F_51(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_30(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_12(x)*F_40(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_66(x))
Eq(F_54(x), F_2(x) + F_55(x))
Eq(F_55(x), F_15(x) + F_56(x) + F_65(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_10(x) + F_30(x))
Eq(F_59(x), F_55(x) + F_60(x))
Eq(F_60(x), 2*F_15(x) + F_51(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_30(x))
Eq(F_64(x), F_60(x))
Eq(F_65(x), F_12(x)*F_54(x))
Eq(F_66(x), F_41(x) + F_67(x))
Eq(F_67(x), 2*F_15(x) + F_47(x) + F_68(x))
Eq(F_68(x), F_12(x)*F_66(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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