0132_1302_2031
Counting sequence:
1, 1, 2, 6, 21, 77, 288, 1093, 4202, 16341, 64187, 254313, 1015163, 4078777, 16481961, 66940960, 273115842, 1118869380, 4600710745, 18982037361, 78561829077, 326080925791, 1357035927107, 5661444867931, 23673369823674, 99202878472677, 416544031136338, 1752333188066763, 7384874688474756, 31174251469597298, 131805916697269545, 558113509272745117, 2366598700957901632, 10048690314212827606, 42721616252213182371, 181849439981013216360, 774955808554836098745, 3306122114475141613211, 14119382013995217181317, 60359636312544911259479, 258280875257145074961425, 1106201819730920641162455, 4741937695754132969573551, 20344197257449590991443829, 87352096250754965399863841, 375352963325514351512741217, 1614085732836742211762875244, 6945776686749791156410742847, 29909588036690046525463292160, 128879553055464961232048925624, 555687679807493880763034327818, 2397399542134222838571874151245, 10349115780573825251714532791930, 44700256989404905425059067687708, 193175468287899920111172205011648, 835258250973229329362730757239091, 3613337248985301506913619211311555, 15638950479851063544655503057490094, 67719058928615198262243683393822026, 293367221596145275719375738575157310, 1271461181813510590653833422640701378, 5512889015863077953485760194653884868, 23913001212650980681595917472488133563, 103767642359390050900255695546741195112, 450461240510806538996252399759123620652, 1956209439995927949682139390170750189728, 8498274771905163997418497732360146553955, 36931670485433868586544323618851046573706, 160551870683841910356693672199431156846051, 698193313798155241803139748106145480269775, 3037216439707739089491195536540626860627467, 13216350737432635200037674684691989337048548, 57528003938160237897893186971229773822723447, 250481310058496945562913240698693276713321389, 1090928024597714482510650450873888483221990147, 4752676380099596082740013084715854336733330991, 20710878173272941794354821165743049631218376999, 90276308118593892354859145411342137318010775479, 393605420667649240784428991860340581538875273359, 1716553996886518085610823169892363633186806792625, 7487902961960005664443135891145059852108470259127, 32671314677499650475617682848460881376667932383937, 142585086020310269469943964595119322232317182696013, 622415212876829476755474908409961106583936000122891, 2717581068822668103475761011569909604003081761252473, 11868032379678387659036025463824899941194670492941633, 51840193580871047276014021734579777237712979874083487, 226487394590244599072930128600960858816688024033882107, 989712172628665785781084777082723363544732684737594819, 4325728307180730455038083693425935440303409110703704317, 18910067554037759638463177581337783554845639037356298026, 82681541692672113612689806472684330082751301361831087803, 361579571981473869576955446605429252957191711561798655108, 1581529480020749394058801766752444280099086141370436246136, 6918741249499054880536939513649719988691066846025908818630, 30272733633405109393490948076538291162232188067163793060719, 132479706456171593142096621196367640604389059127669917068634, 579854063675682671740535589521584375159106745875969151966144, 2538389103583519882580864072234078737591343631527079545287412, 11113897412954276127950473777710083117900734265490640210352548, 48667817765614723546220183612909257141378727268503045964474690
Implicit equation for the generating function in Maple syntax:
(x-1)^2*x^2*F(x)^4+x*(x-1)*(x+1)*F(x)^3+x*(3*x^2-6*x+4)*F(x)^2+(x-1)*(x+1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} x^{2} F \! \left(x \right)^{4}+x \left(x -1\right) \left(x +1\right) F \! \left(x \right)^{3}+x \left(3 x^{2}-6 x +4\right) F \! \left(x \right)^{2}+\left(x -1\right) \left(x +1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 288
a(7) = 1093
a(8) = 4202
a(9) = 16341
a(10) = 64187
a(11) = 254313
a(12) = 1015163
a(13) = 4078777
a(14) = 16481961
a(15) = 66940960
a(n+16) = -50/3*(2*n+3)*(2*n+1)*(n+1)/(2*n+33)/(n+17)/(n+14)*a(n)+2*(2*n+3)*(441*n^2+2173*n+2630)/(2*n+33)/(n+17)/(n+14)*a(n+1)-1/10*(138487*n^3+1266412*n^2+3813521*n+3782340)/(2*n+33)/(n+17)/(n+14)*a(n+2)+1/10*(575855*n^3+6800348*n^2+26642401*n+34632300)/(2*n+33)/(n+17)/(n+14)*a(n+3)-2/15*(1163144*n^3+16935009*n^2+82085491*n+132442818)/(2*n+33)/(n+17)/(n+14)*a(n+4)+1/15*(4504681*n^3+78536592*n^2+456581717*n+884978454)/(2*n+33)/(n+17)/(n+14)*a(n+5)-2/15*(3326038*n^3+67838094*n^2+461606633*n+1047771843)/(2*n+33)/(n+17)/(n+14)*a(n+6)+1/15*(7695121*n^3+179780781*n^2+1401060188*n+3641914452)/(2*n+33)/(n+17)/(n+14)*a(n+7)-1/30*(13999109*n^3+367691094*n^2+3220260577*n+9404266560)/(2*n+33)/(n+17)/(n+14)*a(n+8)+1/15*(4958239*n^3+144173976*n^2+1397347133*n+4514361996)/(2*n+33)/(n+17)/(n+14)*a(n+9)-2/15*(1346029*n^3+42817533*n^2+453818030*n+1602708864)/(2*n+33)/(n+17)/(n+14)*a(n+10)+1/30*(2196607*n^3+75753144*n^2+870091997*n+3328571580)/(2*n+33)/(n+17)/(n+14)*a(n+11)-2/5*(54691*n^3+2030518*n^2+25096826*n+103263207)/(2*n+33)/(n+17)/(n+14)*a(n+12)+1/15*(69182*n^3+2749671*n^2+36364411*n+160009842)/(2*n+33)/(n+17)/(n+14)*a(n+13)-2/15*(4846*n^3+205215*n^2+2890118*n+13534179)/(2*n+33)/(n+17)/(n+14)*a(n+14)+2/15*(403*n^3+18108*n^2+270443*n+1342158)/(2*n+33)/(n+17)/(n+14)*a(n+15), n >= 16
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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 288
a \! \left(7\right) = 1093
a \! \left(8\right) = 4202
a \! \left(9\right) = 16341
a \! \left(10\right) = 64187
a \! \left(11\right) = 254313
a \! \left(12\right) = 1015163
a \! \left(13\right) = 4078777
a \! \left(14\right) = 16481961
a \! \left(15\right) = 66940960
a \! \left(n +16\right) = -\frac{50 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{3 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{2 \left(2 n +3\right) \left(441 n^{2}+2173 n +2630\right) a \! \left(n +1\right)}{\left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{\left(138487 n^{3}+1266412 n^{2}+3813521 n +3782340\right) a \! \left(n +2\right)}{10 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(575855 n^{3}+6800348 n^{2}+26642401 n +34632300\right) a \! \left(n +3\right)}{10 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(1163144 n^{3}+16935009 n^{2}+82085491 n +132442818\right) a \! \left(n +4\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(4504681 n^{3}+78536592 n^{2}+456581717 n +884978454\right) a \! \left(n +5\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(3326038 n^{3}+67838094 n^{2}+461606633 n +1047771843\right) a \! \left(n +6\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(7695121 n^{3}+179780781 n^{2}+1401060188 n +3641914452\right) a \! \left(n +7\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{\left(13999109 n^{3}+367691094 n^{2}+3220260577 n +9404266560\right) a \! \left(n +8\right)}{30 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(4958239 n^{3}+144173976 n^{2}+1397347133 n +4514361996\right) a \! \left(n +9\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(1346029 n^{3}+42817533 n^{2}+453818030 n +1602708864\right) a \! \left(n +10\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(2196607 n^{3}+75753144 n^{2}+870091997 n +3328571580\right) a \! \left(n +11\right)}{30 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(54691 n^{3}+2030518 n^{2}+25096826 n +103263207\right) a \! \left(n +12\right)}{5 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(69182 n^{3}+2749671 n^{2}+36364411 n +160009842\right) a \! \left(n +13\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(4846 n^{3}+205215 n^{2}+2890118 n +13534179\right) a \! \left(n +14\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{2 \left(403 n^{3}+18108 n^{2}+270443 n +1342158\right) a \! \left(n +15\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}, \quad n \geq 16
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/296/
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[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[0,x]*F[4,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[13,x]*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[11,x]*F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]^2*F[4,x]
F[15,x] = F[1,x]+F[13,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_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \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_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_0(x)*F_4(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_13(x)*F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_11(x)*F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)**2*F_4(x))
Eq(F_15(x), F_1(x) + F_13(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": 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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"}}, {"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, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], 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