0132_0231_1032_1203_1230
Counting sequence:
1, 1, 2, 6, 19, 58, 171, 491, 1382, 3831, 10495, 28486, 76755, 205619, 548294, 1456671, 3858583, 10196902, 26895963, 70835051, 186330086, 489662535, 1285803247, 3374300806, 8850730659, 23206202723, 60826597766, 159395225391, 417610736647, 1093957078246, 2865334239915, 7504260234011, 19652009864870, 51461164601559, 134750811292639, 352830997723846, 923823786257523, 2418807864773267, 6332943405445958, 16580726726201151, 43410679882168951, 113654267857805350, 297558171005199867, 779032614663606731, 2039564961753059174, 5339713947642076263, 13979682434289436111, 36599548859505276166, 95819403948877502787, 250859560188615497411, 656761106204317608710, 1719427487967778744911, 4501528957523389813735, 11785174865726109782374, 30854027164852331126859, 80776970805125573627771, 211477015854913583500838, 553654342471993191734007, 1449486551993021276160703, 3794806412385379715149126, 9934934918948533045052691, 26010002884088643809468915, 68095082956689435238129862, 178275264720954111766184031, 466730749252582552086372439, 1221917060282534353151680678, 3199020588392345133899858331, 8375145023100836320037659691, 21926415126546206406047427302, 57404101666256612131482786951, 150285892528554776602576366255, 393453581305856987199435383686, 1030074862309488676631784341475, 2696771027758701931147379579939, 7060238265829098704071983928454, 18483943860634148976309242569071, 48391593500245641056771907112903, 126690837013169726340708450651622, 331680918294842175224496679033323, 868351919401403539782546635685851, 2273374843007240856885630488208806, 5951772615891621720125233672728855, 15581943017361345409443676864392607, 40794056461882088175016666882955206, 106800226420268729359035378296838579, 279606622904100646208562206206997843, 732019642504804153518825975072441926, 1916452305040689406130723705205726591, 5017337273487690654995878146333184567, 13135559517182578552215810812166019366, 34389341281619122614597022580499855675, 90032464334870315769921529777184707211, 235708051737537620156770784981118975590, 617091690907143620633418606695026419879, 1615567021043314363629183288299118246799, 4229609372342879654064813144867543370246, 11073261096227960846265190680183740211843, 28990173916831227140287769484543730457987, 75897260655256072606026421993366750541318, 198701608050937501781276671019793505911951, 520207563501597069026031072283284508661671
Generating function in Maple syntax:
-(x^5+x^4+7*x^3-12*x^2+6*x-1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{x^{5}+x^{4}+7 x^{3}-12 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**5 - x**4 - 7*x**3 + 12*x**2 - 6*x + 1)/((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*F(x)+x^5+x^4+7*x^3-12*x^2+6*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} F \! \left(x \right)+x^{5}+x^{4}+7 x^{3}-12 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/40*(-12*5^(1/2)+60)*(3/2-1/2*5^(1/2))^(-n)+1/40*(12*5^(1/2)+60)*(3/2+1/2*5^(1/2))^(-n)-1/8*2^n*(n+6))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}-\frac{2^{n} \left(n +6\right)}{8} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3), 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) = 19
a \! \left(5\right) = 58
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), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18847/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[39,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[38,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[12,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[14,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[18,x]+F[28,x]+F[35,x]
F[35,x] = F[12,x]*F[23,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[27,x]
F[38,x] = F[12,x]*F[26,x]
F[39,x] = F[18,x]+F[40,x]+F[53,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[10,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[14,x]+F[43,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[18,x]+F[40,x]+F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[23,x]+F[47,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[27,x]+F[50,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[49,x]+F[55,x]
F[55,x] = F[27,x]+F[56,x]
F[56,x] = F[18,x]+F[57,x]+F[58,x]+F[60,x]
F[57,x] = 0
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[47,x]+F[56,x]
F[60,x] = F[12,x]*F[55,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_{22}\! \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_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
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_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{39}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \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_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{53}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{27}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = 0
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_39(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_38(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_12(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_14(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_18(x) + F_28(x) + F_35(x))
Eq(F_35(x), F_12(x)*F_23(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_27(x))
Eq(F_38(x), F_12(x)*F_26(x))
Eq(F_39(x), F_18(x) + F_40(x) + F_53(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_10(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_14(x) + F_43(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_18(x) + F_40(x) + F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_23(x) + F_47(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_27(x) + F_50(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_49(x) + F_55(x))
Eq(F_55(x), F_27(x) + F_56(x))
Eq(F_56(x), F_18(x) + F_57(x) + F_58(x) + F_60(x))
Eq(F_57(x), 0)
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_47(x) + F_56(x))
Eq(F_60(x), F_12(x)*F_55(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": [0, 2, 3, 1], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 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, 1, 3, 2], "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, 0, 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": [1, 2, 3, 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