0231_0312_1230_3012
Counting sequence:
1, 1, 2, 6, 20, 67, 223, 742, 2484, 8399, 28731, 99451, 348127, 1231141, 4393821, 15808046, 57278916, 208844951, 765683811, 2820992111, 10438841451, 38779829501, 144575955621, 540728961251, 2028302456175, 7628647166477, 28762701003653, 108691599851157, 411597600536669, 1561689465974345, 5936123217389625, 22601895971913854, 86193220781648996, 329188507098492903, 1258984925934473459, 4821311304626517415, 18486135500206187059, 70963276939173662989, 272709612340732285013, 1049110733624433669015, 4039923137795383646459, 15571637824952173318117, 60073786970170892461453, 231956623233522692624365, 896359886854956726676069, 3466537508194164833517817, 13416285873844254983828233, 51961016793346383464039635, 201381158071671085519641679, 780986663080789943615900029, 3030682738738123402800965941, 11767889490234402537208973581, 45720160255727114639329548517, 177729025021464037579070525401, 691259896615099112355196714345, 2689974691811063624255263795237, 10472991694858181875067706178813, 40794479318646342156830974805977, 158976466069640136442915047511369, 619809466977734954582975219577769, 2417524903424012966122079774827321, 9433326800836761540044467617832081, 36824263386134294255313312934721137, 143804639587994941472850552269549086, 561792598493034562636774762306264996, 2195524600261172134667030376142848391, 8583312211931737335455486563890976531, 33567667861259679399275610035989757911, 131320534699523127061378308121623674371, 513907647910301943953354769034604864941, 2011751901730670182839867921804380649781, 7877651619687163654608733860413597471311, 30856623704437621612727526095086770251971, 120899846384243221203734291254268168362741, 473833081714512385703796636394996201868221, 1857564939548300262098817703279145400500221, 7284143155029525623697260892945193131233941, 28571020106505631911277496856583066994693401, 112094141097733837707033784288725918035472361, 439893119759200116408661262752295274096001351, 1726695736290471605294671556479767237595224411, 6779321944621927421120951533461000202022489621, 26622938854872311194513802287331522856941558781, 104573733826908040262180516091416037220472089381, 410850477560431003845115443437331993718630253101, 1614497207003740725634625081628971879739692630281, 6345720784641732349810909333385374759185033170361, 24946676330392366762176267799362728203308905421341, 98091298411882221097780028124108751725931755708741, 385773494542002324349016859540912969075729348183081, 1517457431580358634103479528839526155427372414290361, 5970092495944367171695413379905859956266315418653721, 23492230580796619065101300150826713634772530285119241, 92457979874441764474536836918704732929500858349840881, 363948428731620118366526886880056539973312815280473361, 1432877309922591439416743867591575953777575561307621171, 5642224026273709056544752406107458970427427061759032591, 22220934204271039164542512117492704042858302377039125981, 87527297122013860674255846996371231658024381427935107861, 344820066968655713228143533780388460938069713832654652061, 1358651250233213573481637597239188700340935266602514166101
Generating function in Maple syntax:
(-(x-1)^2*(-4*x+1)^(1/2)-x^2-2*x+1)/(4*x^3-6*x^2+2*x)
Generating function in latex syntax:
\frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}-x^{2}-2 x +1}{4 x^{3}-6 x^{2}+2 x}
Generating function in sympy syntax:
(-x**2 - 2*x - sqrt(1 - 4*x)*(x - 1)**2 + 1)/(4*x**3 - 6*x**2 + 2*x)
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^2*F(x)^2+(2*x-1)*(x-1)*(x^2+2*x-1)*F(x)+x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 4*(1+2*n)/(4+n)*a(n)-2*(13+7*n)/(4+n)*a(1+n)+(20+7*n)/(4+n)*a(n+2)+6/(4+n), n >= 4
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(n +3\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{4+n}-\frac{2 \left(13+7 n \right) a \! \left(1+n \right)}{4+n}+\frac{\left(20+7 n \right) a \! \left(n +2\right)}{4+n}+\frac{6}{4+n}, \quad n \geq 4
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21412/
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[9,x]
F[3,x] = F[1,x]+F[31,x]+F[4,x]
F[4,x] = F[5,x]*F[9,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[11,x,k[0]]+F[28,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]*F[9,x]
F[8,x,k[0]] = 1/(-1+k[0])*(F[6,x,k[0]]*k[0]-F[6,x,1])
F[9,x] = F[10,x,1]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[10,x,k[0]]*F[13,x,k[0]]*F[17,x]
F[13,x,k[0]] = F[1,x]+F[14,x,k[0]]+F[16,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]*F[9,x]
F[15,x,k[0]] = 1/(-1+k[0])*(F[13,x,k[0]]*k[0]-F[13,x,1])
F[16,x,k[0]] = F[10,x,k[0]]*F[13,x,k[0]]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[9,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[9,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[9,x]
F[27,x] = F[18,x]+F[25,x]
F[28,x,k[0]] = F[29,x,k[0]]
F[29,x,k[0]] = F[17,x]*F[30,x,k[0]]*F[9,x]
F[30,x,k[0]] = 1/(-1+k[0])*(F[13,x,k[0]]*k[0]-F[13,x,1])
F[31,x] = F[32,x]
F[32,x] = F[17,x]*F[33,x]*F[9,x]
F[33,x] = F[13,x,1]
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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{8}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{9}\! \left(x \right) = F_{10}\! \left(x , 1\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{17}\! \left(x \right)
F_{13}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{15}\! \left(x , y\right) = \frac{y F_{13}\! \left(x , y\right)-F_{13}\! \left(x , 1\right)}{-1+y}
F_{16}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{17}\! \left(x \right) F_{30}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{30}\! \left(x , y\right) = \frac{y F_{13}\! \left(x , y\right)-F_{13}\! \left(x , 1\right)}{-1+y}
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right) F_{33}\! \left(x \right) F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x , 1\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)*F_9(x))
Eq(F_3(x), F_1(x) + F_31(x) + F_4(x))
Eq(F_4(x), F_5(x)*F_9(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_11(x, k_0) + F_28(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0)*F_9(x))
Eq(F_8(x, k_0), (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_9(x), F_10(x, 1))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_10(x, k_0)*F_13(x, k_0)*F_17(x))
Eq(F_13(x, k_0), F_1(x) + F_14(x, k_0) + F_16(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0)*F_9(x))
Eq(F_15(x, k_0), (k_0*F_13(x, k_0) - F_13(x, 1))/(k_0 - 1))
Eq(F_16(x, k_0), F_10(x, k_0)*F_13(x, k_0))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_9(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_9(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_9(x))
Eq(F_27(x), F_18(x) + F_25(x))
Eq(F_28(x, k_0), F_29(x, k_0))
Eq(F_29(x, k_0), F_17(x)*F_30(x, k_0)*F_9(x))
Eq(F_30(x, k_0), (-k_0*F_13(x, k_0) + F_13(x, 1))/(1 - k_0))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_17(x)*F_33(x)*F_9(x))
Eq(F_33(x), F_13(x, 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [3, 0, 1, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], 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