0213_1032_1320
Counting sequence:
1, 1, 2, 6, 21, 75, 265, 927, 3229, 11253, 39355, 138362, 489440, 1742576, 6244395, 22516585, 81673947, 297894234, 1092086024, 4022418903, 14879231036, 55255415912, 205931156743, 769994355746, 2887687533958, 10859294097589, 40939724979676, 154701255400490, 585829636549849, 2222842322625552, 8449758084172550, 32175331228072879, 122714337057405492, 468723122442229346, 1792857530206173267, 6866679797488012117, 26332131539567105593, 101095760348890235016, 388561407469602773354, 1494997815649095640420, 5757740024327760035475, 22195881483664214803013, 85640952367275849777691, 330720178063905907804456, 1278182042810100334733922, 4943806328694266633977744, 19136032630995818948899653, 74122501148158998289953147, 287304769877423721133211609, 1114340573242540679922006534, 4324778554547977375693149872, 16794605809199257612625921180, 65256772353253639503059416069, 253700670657369485885274441791, 986844793876352869253069821879, 3840597306414832683387642402076, 14954204135345261233118948198394, 58255215952206264990416889411214, 227041934057638513123664738085579, 885258758307032311662018119105601, 3453192371315812667906410306264001, 13475705708252330212296801047825406, 52608581261261797011147972503338232, 205461472041366372756021426742846766, 802726511635244151213338647325364853, 3137351264941590689468615817124288931, 12266260099468935425854634599881041641, 47974455324223666979528567793057173281, 187694859354428420674122998973715276329, 734572934689557645028560196051867982821, 2875766672264687789280747358077506651937, 11261718863538014477067214387721330669331, 44114800925265422816338356379897754481335, 172857816407203283381996022454060193570391, 677509474197353118440321190243077618130473, 2656195927889861789147739621688053144798971, 10416458471850798212012779300906562240846071, 40859429574463700890841539391625430008498341, 160314884746411081160432160237263990231849719, 629161217643397242982594267968919280820717211, 2469755208688592383856816194968184081486133317, 9697218109595442534111031600763613669847690151, 38083707454656191192022581100143308046823062515, 149598653589119303875363919881750651973869766691, 587773833826533100873729649431713433345017275153, 2309854823083530299582178394908661373117982947401, 9079226395770335236034787010172523221502025962681, 35694446923066586859048458733103782795112419924851, 140358305560198531756782467723589699358973821030659, 552025690791717463848018721510176063799815823272691, 2171512367025703003150213201622583094455189673746157, 8543687611849737784649845853610691839627304189049841, 33620696187544618698658210114663359262992334304306945, 132325834122269163707411605155609965358790882623915091, 520903814377960255503737080454714023384954541553832143, 2050896807230703229035879995579319448066611500874183971, 8076104298420262639687665772424137592027838402843884521, 31807561820252402198399443251997740867740540001143829661, 125293271745776348469748173202604778374000551405968936849, 493619867604981256348321530660465968518595522883156701521, 1945018192568497814496508611857177593539243854919275051337
Generating function in Maple syntax:
1/2*((-x^4+8*x^3-16*x^2+10*x-2)*(1-4*x)^(1/2)+3*x^4-18*x^3+24*x^2-12*x+2)/x/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{\left(-x^{4}+8 x^{3}-16 x^{2}+10 x -2\right) \sqrt{1-4 x}+3 x^{4}-18 x^{3}+24 x^{2}-12 x +2}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(3*x**4/2 - 9*x**3 + 12*x**2 - 6*x + sqrt(1 - 4*x)*(-x**4 + 8*x**3 - 16*x**2 + 10*x - 2)/2 + 1)/(x*(x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2-3*x+1)^2*(x-1)^4*F(x)^2-(x^2-3*x+1)*(3*x^4-18*x^3+24*x^2-12*x+2)*(x-1)^2*F(x)+x^8-14*x^7+73*x^6-183*x^5+255*x^4-202*x^3+90*x^2-21*x+2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}-\left(x^{2}-3 x +1\right) \left(3 x^{4}-18 x^{3}+24 x^{2}-12 x +2\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-14 x^{7}+73 x^{6}-183 x^{5}+255 x^{4}-202 x^{3}+90 x^{2}-21 x +2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 265
a(7) = 927
a(8) = 3229
a(n+8) = -(1+2*n)/(n+9)*a(n)+1/2*(86+49*n)/(n+9)*a(1+n)-1/2*(611+220*n)/(n+9)*a(n+2)+3/2*(597+160*n)/(n+9)*a(n+3)-1/2*(2629+563*n)/(n+9)*a(n+4)+(1049+185*n)/(n+9)*a(n+5)-2*(229+34*n)/(n+9)*a(n+6)+(102+13*n)/(n+9)*a(n+7)+5/2/(n+9), 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) = 21
a \! \left(5\right) = 75
a \! \left(6\right) = 265
a \! \left(7\right) = 927
a \! \left(8\right) = 3229
a \! \left(n +8\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{n +9}+\frac{\left(86+49 n \right) a \! \left(1+n \right)}{2 n +18}-\frac{\left(611+220 n \right) a \! \left(n +2\right)}{2 \left(n +9\right)}+\frac{3 \left(597+160 n \right) a \! \left(n +3\right)}{2 \left(n +9\right)}-\frac{\left(2629+563 n \right) a \! \left(n +4\right)}{2 \left(n +9\right)}+\frac{\left(1049+185 n \right) a \! \left(n +5\right)}{n +9}-\frac{2 \left(229+34 n \right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{5}{2 \left(n +9\right)}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/324/
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[11,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[11,x]*F[12,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[7,x]^2*F[11,x]
F[11,x] = x
F[12,x] = F[13,x]+F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[11,x]*F[15,x]
F[17,x] = F[18,x]+F[23,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[20,x]
F[20,x] = F[15,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[15,x]^2*F[11,x]*F[20,x]
F[23,x] = F[24,x]*F[9,x]
F[24,x] = F[16,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_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{7} \left(x \right)^{2} F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \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) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{11}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \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_11(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_11(x)*F_12(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_7(x)**2)
Eq(F_11(x), x)
Eq(F_12(x), F_13(x) + F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_11(x)*F_15(x))
Eq(F_17(x), F_18(x) + F_23(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_20(x))
Eq(F_20(x), F_15(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_15(x)**2*F_20(x))
Eq(F_23(x), F_24(x)*F_9(x))
Eq(F_24(x), F_16(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 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": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 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": [1, 0, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], 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