0132_0213_0231_1023_2013
Counting sequence:
1, 1, 2, 6, 19, 61, 201, 678, 2328, 8107, 28564, 101642, 364755, 1318581, 4797204, 17551553, 64537871, 238371805, 883976307, 3290064052, 12285747283, 46015779590, 172826466138, 650751983903, 2456050324001, 9289659257421, 35207524053734, 133685346101493, 508499200796894, 1937339450430164, 7392402535880032, 28248114086641990, 108088724451222047, 414118563392925545, 1588515630650132042, 6100316971446251149, 23452076548896344054, 90251430168133229292, 347655000713434538519, 1340428999651846698431, 5172743271205469729391, 19978458702866274850370, 77223658740849028178229, 298724338765485519577520, 1156400313805987957847390, 4479706861112564077415221, 17365291936497486694589889, 67358709023521817769415201, 261440814513228067204243900, 1015337536870859999666886563, 3945435821919316990641236886, 15339727349462865172075402538, 59671858619113702165026229883, 232242871829400989524423745575, 904332647520364440637369227783, 3523055766605825784554166508726, 13731222647876210239400088184687, 53541500548520961767082044724698, 208860877565449217700639420194868, 815084806209302725075000557000341, 3182158740296053079300631600803345, 12428213571836741688368555600036971, 48557703636324585574887252123734517, 189786414658215350467797375819674998, 742035809318043799121176195990869568, 2902236471557056658239740286040964227, 11354932196880517652110570522748902800, 44440192726103604874952565088620687782, 173981455784534589160180523077711180691, 681336718955963347550917229148490782409, 2669001180678157938694524697029325764009, 10458282148193871362318857545481745141200, 40991440778116711518314185742269547594041, 160710411050409966738550163308775744089190, 630245461850027770810004774070619963824600, 2472221265467266412935667202426034193362259, 9700049885599607543932789317181623726952631, 38068595171003311425989808677907473417577049, 149438783489862829410784138582343469963273822, 586760436532684946208324376260322590241134267, 2304394761419499346984343954216833395013471074, 9052095185833785168759550920913880569133572864, 35566016554425724726430637686201688115246158979, 139769691982810530350661949415152118623298245487, 549389457091934635332987347494754090028991829769, 2159907249933858444014090953219082823450746179932, 8493278393456519820244713501373597174906996199661, 33404053331748027970407303347466995935271084856050, 131402814610598586126689851389853525374430842429318, 516999423349812497186424122452715127199766683576061, 2034480823876876584030159838215079122391718367642829, 8007438463215230533246557636184800045076688618776053, 31521614661246899195691643552631253703349131581611086, 124107080563380411518747519067315964937456710837509685, 488715814109300028588329130892180995563791059948825712, 1924803820986593115485588555697104431234821945156097802, 7582027625199164167856249310832814039512763248784449963, 29871131280814826457864682132855133283978606817378329525, 117702055010779460271869366818718448135343872805670178763, 463853861566645049781898589825365412472243311951058203014, 1828276139456527937952255197179730328551906280195138731261
Generating function in Maple syntax:
1/2*(-(x^2-x+1)^2*(1-4*x)^(1/2)+3*x^4-4*x^3+7*x^2-4*x+1)/(x^3+x+1)/x^2
Generating function in latex syntax:
\frac{-\left(x^{2}-x +1\right)^{2} \sqrt{1-4 x}+3 x^{4}-4 x^{3}+7 x^{2}-4 x +1}{2 \left(x^{3}+x +1\right) x^{2}}
Generating function in sympy syntax:
(3*x**4/2 - 2*x**3 + 7*x**2/2 - 2*x - sqrt(1 - 4*x)*(x**2 - x + 1)**2/2 + 1/2)/(x**2*(x**3 + x + 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x+1)*x^2*F(x)^2+(-3*x^4+4*x^3-7*x^2+4*x-1)*F(x)+x^4-2*x^3+4*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x +1\right) x^{2} F \! \left(x \right)^{2}+\left(-3 x^{4}+4 x^{3}-7 x^{2}+4 x -1\right) F \! \left(x \right)+x^{4}-2 x^{3}+4 x^{2}-3 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 61
a(n+6) = 2*(1+2*n)/(8+n)*a(n)+3*(11+3*n)/(8+n)*a(2+n)-(16+5*n)/(8+n)*a(n+1)-2*(11+n)/(8+n)*a(n+3)+15/(8+n)*a(n+4)+(23+4*n)/(8+n)*a(n+5), 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) = 61
a \! \left(n +6\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{3 \left(11+3 n \right) a \! \left(2+n \right)}{8+n}-\frac{\left(16+5 n \right) a \! \left(n +1\right)}{8+n}-\frac{2 \left(11+n \right) a \! \left(n +3\right)}{8+n}+\frac{15 a \! \left(n +4\right)}{8+n}+\frac{\left(23+4 n \right) a \! \left(n +5\right)}{8+n}, \quad n \geq 6
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21558/
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[23,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[10,x]*F[6,x]*F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[6,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x,1]
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[14,x,k[0]]
F[12,x,k[0]] = F[11,x,k[0]]*F[13,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x,k[0]] = F[15,x,k[0]]*F[9,x]
F[15,x,k[0]] = F[1,x]+F[16,x,k[0]]+F[18,x,k[0]]+F[21,x,k[0]]
F[16,x,k[0]] = F[13,x,k[0]]*F[17,x,k[0]]
F[17,x,k[0]] = (F[11,x,k[0]]*k[0]-F[11,x,1])/(-1+k[0])
F[18,x,k[0]] = F[19,x,k[0]]
F[19,x,k[0]] = F[10,x]*F[20,x,k[0]]*F[6,x]*F[9,x]
F[20,x,k[0]] = F[1,x]+F[13,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]*F[9,x]
F[22,x,k[0]] = (F[15,x,k[0]]*k[0]-F[15,x,1])/(-1+k[0])
F[23,x] = F[14,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_{23}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \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_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x , 1\right)
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{10}\! \left(x \right) F_{20}\! \left(x , y\right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{22}\! \left(x , y\right) = \frac{y F_{15}\! \left(x , y\right)-F_{15}\! \left(x , 1\right)}{-1+y}
F_{23}\! \left(x \right) = F_{14}\! \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_23(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_10(x)*F_6(x)*F_9(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_6(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x, 1))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0) + F_14(x, k_0))
Eq(F_12(x, k_0), F_11(x, k_0)*F_13(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x, k_0), F_15(x, k_0)*F_9(x))
Eq(F_15(x, k_0), F_1(x) + F_16(x, k_0) + F_18(x, k_0) + F_21(x, k_0))
Eq(F_16(x, k_0), F_13(x, k_0)*F_17(x, k_0))
Eq(F_17(x, k_0), (-k_0*F_11(x, k_0) + F_11(x, 1))/(1 - k_0))
Eq(F_18(x, k_0), F_19(x, k_0))
Eq(F_19(x, k_0), F_10(x)*F_20(x, k_0)*F_6(x)*F_9(x))
Eq(F_20(x, k_0), F_1(x) + F_13(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0)*F_9(x))
Eq(F_22(x, k_0), (-k_0*F_15(x, k_0) + F_15(x, 1))/(1 - k_0))
Eq(F_23(x), F_14(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, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "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], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 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