0213_1032_1230_1320_3102
Counting sequence:
1, 1, 2, 6, 19, 50, 122, 288, 672, 1567, 3670, 8647, 20493, 48809, 116703, 279848, 672465, 1618292, 3898416, 9397762, 22665670, 54683161, 131957436, 318477153, 768715883, 1855591715, 4479385741, 10813531866, 26105103967, 63021562310, 152144704886, 367305270132, 886746018684, 2140782378211, 5168286618418, 12477316528075, 30122856429849, 72722927054957, 175568544961035, 423859749064220, 1023287609596605, 2470434266850344, 5964155008395852, 14398742447331958, 34761636931846594, 83922011503500157, 202605652160106936, 489133303237447173, 1180872238269992471, 2850877746826154375, 6882627678606012505, 16616133017770610670, 40114893574563374059, 96845919941045927834, 233806733091219936434, 564459385532199073800, 1362725503198896061080, 3289910390381982443239, 7942546281458130168910, 19175002949245503246607, 46292552173391666345829, 111760107285418626084209, 269812766727061238340471, 651385640711763212733776, 1572584048105642093599209, 3796553736850323939688220, 9165691521688620942518952, 22127936780037173334021418, 53421565081454906089396222, 128971066942448531500939297, 311363698965545453558230260, 751698464872234469072476137, 1814760628707902906625209603, 4381219722284623827699993899, 10577200073271622622324318053, 25535619868818924678024844098, 61648439810894999644349335735, 148832499490585507238375053742, 359313438792028125058726305326, 867459377074580451565106058972, 2094232192941089833335843674100, 5055923762956599617592977046571, 12206079718854029373024886651306, 29468083200664238167502022866323, 71142246120181825816391293778673, 171752575441026789712506244328885, 414647397002233625261987777729571, 1001047369445491160169287428979396, 2416742135893211285553952260165717, 5834531641231906191163387202972240, 14085805418357011467720311544241476, 34006142477945909386329790423240222, 82098090374248798299945257400630394, 198202323226443454305511450366186501, 478502736827135623289824668284589168, 1155207796880714565583308442228627725, 2788918330588564535533445718186684719, 6733044458057843282425351699340091407, 16255007246704250527236305103049792945, 39243058951466343409525269712360687830, 94741125149636935845766308320875095235
Generating function in Maple syntax:
(x^9+2*x^8-4*x^7-4*x^6+5*x^5+5*x^4+x^3-8*x^2+5*x-1)/(x^2+x-1)/(x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{9}+2 x^{8}-4 x^{7}-4 x^{6}+5 x^{5}+5 x^{4}+x^{3}-8 x^{2}+5 x -1}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**9 + 2*x**8 - 4*x**7 - 4*x**6 + 5*x**5 + 5*x**4 + x**3 - 8*x**2 + 5*x - 1)/((x - 1)**3*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^2+2*x-1)*(x-1)^3*F(x)+x^9+2*x^8-4*x^7-4*x^6+5*x^5+5*x^4+x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+2 x^{8}-4 x^{7}-4 x^{6}+5 x^{5}+5 x^{4}+x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),2,n = 2,1,0)+1/10*(-7*5^(1/2)+15)*(-1/2*5^(1/2)-1/2)^(-n)+1/10*(7*5^(1/2)+15)*(1/2*5^(1/2)-1/2)^(-n)-1/2*n^2-3/2*n+1/2*(-1-2^(1/2))^(-n)+1/2*(2^(1/2)-1)^(-n)-5
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}2 & n =0\text{ or } n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(-7 \sqrt{5}+15\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(7 \sqrt{5}+15\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}-\frac{n^{2}}{2}-\frac{3 n}{2}+\frac{\left(-1-\sqrt{2}\right)^{-n}}{2}+\frac{\left(\sqrt{2}-1\right)^{-n}}{2}-5
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 122
a(7) = 288
a(8) = 672
a(9) = 1567
a(n+1) = -1/3*n^2-1/3*a(n)+a(n+3)-1/3*a(n+4)-1/3*n-1, n >= 10
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) = 50
a \! \left(6\right) = 122
a \! \left(7\right) = 288
a \! \left(8\right) = 672
a \! \left(9\right) = 1567
a \! \left(n +1\right) = -\frac{n^{2}}{3}-\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}-\frac{n}{3}-1, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20563/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[7,x]
F[15,x] = F[16,x]+F[17,x]+F[47,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[29,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[16,x]+F[24,x]+F[27,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[19,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[7,x]
F[29,x] = F[30,x]+F[44,x]
F[30,x] = F[16,x]+F[17,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[7,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[20,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[36,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[33,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[33,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[34,x]
F[51,x] = F[38,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{47}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{20}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{33}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{34}\! \left(x \right)
F_{51}\! \left(x \right) = F_{38}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_14(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_7(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_47(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_29(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_16(x) + F_24(x) + F_27(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_19(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_7(x))
Eq(F_29(x), F_30(x) + F_44(x))
Eq(F_30(x), F_16(x) + F_17(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_7(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_20(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_33(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_33(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_34(x))
Eq(F_51(x), F_38(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, 2, 1, 3], "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, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 2, 1, 3], "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, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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