0132_0213_0321_1320_2301
Counting sequence:
1, 1, 2, 6, 19, 54, 140, 346, 842, 2055, 5070, 12665, 31991, 81529, 209185, 539426, 1396209, 3623864, 9425278, 24552492, 64034164, 167155073, 436642592, 1141197451, 2983801249, 7803911619, 20415346995, 53416959316, 139785194507, 365837955490, 957527339280, 2506341402350, 6560691553806, 17174122637899, 44958455125090, 117694800276141, 308113060790139, 806618612278109, 2111691236423093, 5528352017761302, 14473158658414549, 37890711640604508, 99198151629659282, 259702093980911296, 679904831778169304, 1780005804283806309, 4660099386933690900, 12200265968238172511, 31940645741222664197, 83621565702313522375, 218923840259485336615, 573149532863677385960, 1500523913906616651471, 3928420520006312264294, 10284734268412599570820, 26925775529832045347106, 70492578810284654310994, 184551933879424153312079, 483163168784792277125558, 1264937464388561621115937, 3311649008208110472378559, 8670009127890205568388993, 22698377510771377777586697, 59425121675041670854030666, 155576984055589120963890809, 407305823574196664396482624, 1066340472831942816943312230, 2791715567251515675869041556, 7308806173582371989535068412, 19134702842815135850478760585, 50095302133502106677386496200, 131151203114969326412651386995, 343358306325962157022509080361, 898923714092029713578758787339, 2353412832408352121561533254875, 6161314776049476926801373035196, 16130531481572979210233650082899, 42230279640335261806681705799274, 110560307382764408415375724613048, 289450642394621167850573982768022, 757791619574425503958603253284694, 1983924215875308161669749836416403, 5193981027144804616339674374773962, 13598018863745716957927329525679061, 35600075560465568798598426677971139, 93202207810397434520180175459817189, 244006547856219624926566549604819485, 638817435729247220588768373161499326, 1672445759273493597498236369493581277, 4378519841975176693222936334547243620, 11463113766419922724804563832604348410, 30010821456820363966458737560178407080, 78569350603112714145107613641756298976, 205697230350660868409936032952741564909, 538522340445156070966844344391770776380, 1409869790977379704254884718573175763911, 3691087032472127761326385248028966761229, 9663391306409293018781421898916145266319, 25299086886696330173132182195524310794063, 66233869353560855256843728181266470900432, 173402521173748551109856209335494469757079
Generating function in Maple syntax:
(x^8-x^7-9*x^6+21*x^5-38*x^4+43*x^3-26*x^2+8*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{8}-x^{7}-9 x^{6}+21 x^{5}-38 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**8 - x**7 - 9*x**6 + 21*x**5 - 38*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)*(x-1)^4*F(x)+x^8-x^7-9*x^6+21*x^5-38*x^4+43*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-x^{7}-9 x^{6}+21 x^{5}-38 x^{4}+43 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/60*(-6*5^(1/2)+30)*(3/2-1/2*5^(1/2))^(-n)+1/60*(6*5^(1/2)+30)*(3/2+1/2*5^(1/2))^(-n)+1/3*n^3-5/2*n^2+25/6*n+3/4*2^n-4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-6 \sqrt{5}+30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(6 \sqrt{5}+30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{n^{3}}{3}-\frac{5 n^{2}}{2}+\\\frac{25 n}{6}+\frac{3 \,2^{n}}{4}-4 & \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) = 54
a(6) = 140
a(7) = 346
a(8) = 842
a(n+3) = 1/3*n^3-5/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)+1/6*n+4, 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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 140
a \! \left(7\right) = 346
a \! \left(8\right) = 842
a \! \left(n +3\right) = \frac{n^{3}}{3}-\frac{5 n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{n}{6}+4, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21605/
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[14,x]*F[4,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[14,x]*F[9,x]
F[9,x] = F[10,x]+F[15,x]
F[10,x] = F[11,x]*F[5,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[16,x]
F[16,x] = F[11,x]^2*F[14,x]*F[7,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]*F[19,x]
F[19,x] = F[20,x]+F[38,x]
F[20,x] = F[21,x]+F[9,x]
F[21,x] = F[11,x]*F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[24,x]*F[35,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[12,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[32,x]
F[27,x] = 0
F[28,x] = F[14,x]*F[29,x]
F[29,x] = F[14,x]+F[30,x]
F[30,x] = F[27,x]+F[28,x]+F[31,x]
F[31,x] = F[12,x]*F[14,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[12,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[1,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[14,x]*F[35,x]
F[38,x] = F[39,x]
F[39,x] = F[14,x]*F[40,x]*F[46,x]
F[40,x] = F[35,x]+F[41,x]
F[41,x] = F[12,x]+F[42,x]
F[42,x] = F[27,x]+F[43,x]+F[45,x]
F[43,x] = F[14,x]*F[44,x]
F[44,x] = F[36,x]+F[42,x]
F[45,x] = F[14,x]*F[41,x]
F[46,x] = F[15,x]+F[47,x]
F[47,x] = F[11,x]^3
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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \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_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \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_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{24}\! \left(x \right) F_{35}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{40}\! \left(x \right) F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \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_{27}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11} \left(x \right)^{3}
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_14(x)*F_4(x))
Eq(F_4(x), F_17(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_8(x))
Eq(F_8(x), F_14(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_15(x))
Eq(F_10(x), F_11(x)*F_5(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_11(x)**2*F_14(x)*F_7(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_14(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_38(x))
Eq(F_20(x), F_21(x) + F_9(x))
Eq(F_21(x), F_11(x)*F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_24(x)*F_35(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_12(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_32(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_14(x)*F_29(x))
Eq(F_29(x), F_14(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_28(x) + F_31(x))
Eq(F_31(x), F_12(x)*F_14(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_12(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_1(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_14(x)*F_35(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_14(x)*F_40(x)*F_46(x))
Eq(F_40(x), F_35(x) + F_41(x))
Eq(F_41(x), F_12(x) + F_42(x))
Eq(F_42(x), F_27(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_14(x)*F_44(x))
Eq(F_44(x), F_36(x) + F_42(x))
Eq(F_45(x), F_14(x)*F_41(x))
Eq(F_46(x), F_15(x) + F_47(x))
Eq(F_47(x), F_11(x)**3)
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": 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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], 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