0123_0132_2103_2130
Counting sequence:
1, 1, 2, 6, 20, 60, 166, 446, 1160, 2958, 7428, 18452, 45474, 111426, 271892, 661446, 1605628, 3891468, 9420902, 22788790, 55093480, 133137822, 321644892, 776895700, 1876229106, 4530694026, 10939878244, 26414259078, 63774801572, 153974618844, 371742079462, 897488995886, 2166770629256, 5231114751534, 12629141208180, 30489632482580, 73608798326274, 177707882106834, 429025648931060, 1035760986102342, 2500550621697292, 6036867210999468, 14574293308604966, 35185467532272934, 84945251082620008, 205076007309194814, 495097327960806732, 1195270766239082452, 2885639030782484178, 6966549109368074010, 16818737714713845700, 40604025307054916358, 98026789597065634868, 236657606593974799740, 571342006237120902694, 1379341624760473491614, 3330025265140868143496, 8039392170503067229902, 19408809631614898322148, 46857011475671655361556, 113122832651999172578658, 273102676893294070184034, 659328186625530937116884, 1591759050451846543869126, 3842846288034859745064220, 9277451627352816755861580, 22397749544106705597651686, 54072950717811141314888086, 130543650983416890131801320, 315160252690701963347368158, 760864156374766763200355388, 1836888565456563707092121812, 4434641287314694573004374194, 10706171140129932927167950506, 25846983567646718590032307492, 62400138275541737498100117126, 150647260118924325986905741508, 363694658513708725264674406428, 878036577146863687558026231142, 2119767812808291626064819077390, 5117572202764849098111421667720, 12354912218340287506942963025262, 29827396639449188677321750130772, 72009705497244831834358961952020, 173846807633948953329174715271106, 419703320765159280616446868194546, 1013253449164294601281811044554164, 2446210219093792829804250037653318, 5905673887351952855627803880818732, 14257557993797817361595800740068844, 34420789874947782034660679249875622, 83099137743693699627036483356170246, 200619065362335701901004189441321576, 484337268468365955117486782594758014, 1169293602299069005277052216700212780, 2822924473066506244221521649415317460, 6815142548432085219971876460249495954, 16453209569930682777248176048277044794, 39721561688293460736644877129338061700, 95896332946517620536920636607527425542, 231514227581328728433010627616631784532
Generating function in Maple syntax:
-(2*x^8+x^7-7*x^6-2*x^5+4*x^4+6*x^3-3*x+1)/(x^2+2*x-1)/(x^2+x-1)^2
Generating function in latex syntax:
-\frac{2 x^{8}+x^{7}-7 x^{6}-2 x^{5}+4 x^{4}+6 x^{3}-3 x +1}{\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**8 - x**7 + 7*x**6 + 2*x**5 - 4*x**4 - 6*x**3 + 3*x - 1)/((x**2 + x - 1)**2*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(x^2+x-1)^2*F(x)+2*x^8+x^7-7*x^6-2*x^5+4*x^4+6*x^3-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{8}+x^{7}-7 x^{6}-2 x^{5}+4 x^{4}+6 x^{3}-3 x +1 = 0
Explicit closed form in Maple syntax:
1/2*piecewise(n = 0,2,n = 1,2,n = 2,4,-11*(-1-2^(1/2))^(-n-1)-11*(2^(1/2)-1)^(-n-1)+1/5*((-2*n+4)*5^(1/2)-10*n)*(-1/2*5^(1/2)-1/2)^(-n)+1/5*((2*n-4)*5^(1/2)-10*n)*(1/2*5^(1/2)-1/2)^(-n)+29*(-1-2^(1/2))^(-n)+29*(2^(1/2)-1)^(-n))
Explicit closed form in latex syntax:
\frac{\left(\left\{\begin{array}{cc}2 & n =0 \\ 2 & n =1 \\ 4 & n =2 \\ -11 \left(-1-\sqrt{2}\right)^{-n -1}-11 \left(\sqrt{2}-1\right)^{-n -1}+\\\frac{\left(\left(-2 n +4\right) \sqrt{5}-10 n \right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\\\frac{\left(\left(2 n -4\right) \sqrt{5}-10 n \right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+29 \left(-1-\sqrt{2}\right)^{-n}+29 \left(\sqrt{2}-1\right)^{-n} & \mathit{\text{otherwise}} \end{array}\right.\right)}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 166
a(7) = 446
a(8) = 1160
a(n+6) = a(n)+4*a(n+1)+2*a(n+2)-6*a(n+3)-2*a(n+4)+4*a(n+5), 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) = 20
a \! \left(5\right) = 60
a \! \left(6\right) = 166
a \! \left(7\right) = 446
a \! \left(8\right) = 1160
a \! \left(n +6\right) = a \! \left(n \right)+4 a \! \left(n +1\right)+2 a \! \left(n +2\right)-6 a \! \left(n +3\right)-2 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14876/
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[17,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[16,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[4,x]*F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[13,x]+F[19,x]+F[28,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[13,x]+F[14,x]+F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[22,x]*F[4,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[36,x]
F[30,x] = F[2,x]+F[31,x]
F[31,x] = F[13,x]+F[32,x]+F[35,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]+F[4,x]
F[35,x] = F[2,x]*F[4,x]
F[36,x] = F[18,x]+F[37,x]
F[37,x] = F[13,x]+F[38,x]+F[41,x]+F[42,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]+F[26,x]
F[41,x] = 0
F[42,x] = F[18,x]*F[4,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_{17}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{28}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \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_17(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_16(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_4(x)*F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_13(x) + F_19(x) + F_28(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_13(x) + F_14(x) + F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_22(x)*F_4(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_36(x))
Eq(F_30(x), F_2(x) + F_31(x))
Eq(F_31(x), F_13(x) + F_32(x) + F_35(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x) + F_4(x))
Eq(F_35(x), F_2(x)*F_4(x))
Eq(F_36(x), F_18(x) + F_37(x))
Eq(F_37(x), F_13(x) + F_38(x) + F_41(x) + F_42(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x) + F_26(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_18(x)*F_4(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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