0231_1203_1230_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 60, 191, 617, 2026, 6762, 22922, 78822, 274564, 967396, 3442872, 12360601, 44717010, 162854575, 596572801, 2196650053, 8125261292, 30177113201, 112486327722, 420676621426, 1577941052309, 5934886478587, 22377620648687, 84568456797776, 320272427391893, 1215289650304962, 4619862709402152, 17591936486142166, 67094247738920171, 256271590350565272, 980211421564916759, 3754113026550330006, 14395625724047220660, 55266184150218473637, 212405973504926843119, 817197575895351272533, 3147148675949381674484, 12131536566773711828835, 46806139816407973493320, 180742313193635100572994, 698505249636679138410817, 2701570765340784792592243, 10456466138817965099416669, 40500603200639279094439988, 156975893340741188792667539, 608817482772911982178414332, 2362721108024519137051842856, 9174833786410991108636694047, 35647919288275388651251494601, 138583277372280047775728205921, 539037646154110087294462730772, 2097734212952578688434353742903, 8167644984568290803154357491874, 31816370378992559673014903358808, 123995113266517226043589500426631, 483450259928824605433227543258703, 1885757330856413069576610716742791, 7358688175868211399492161085526502, 28726961891772475681336380264726507, 112188436895230963534176908410039609, 438298778548068895581200308452202490, 1712975301202257238330140951197346322, 6697084307582096614387736911409483370, 26192057916742365867599395276813523706, 102470355549885913484572093003652779300, 401021259118399463366213838969017777808, 1569903772036727612531885892378661344580, 6147680336340216213241081494327453105574, 24081216162587306898610907261579128291120, 94356297448585428542070565020841453418642, 369815651817275169685510596166223741513750, 1449834173919833540947849127498895340332186, 5685476628213935773428101434480136193236380, 22301183209319421363649065173016870709080818, 87498077113212563741251307698150559992969900, 343380711139946658159402982277314625419095474, 1347899084932063332731995829809709224379621450, 5292250225690083152189936885298604768344297312, 20783676736664628752951253388935478170328464160, 81639625501493272666346603722454346526304272846, 320755365894757187572047728716194146724004897770, 1260488611604761935312439515854670270543383558998, 4954431605301373089433194885727391999086914890280, 19477651230312631568662278341280454180727957835714, 76588781691453844809293138198568017763695341900200, 301215715196189453449144320173391390598893811033552, 1184873826246949010969468658162011171797514696395030, 4661726475549101785650429978341921428442060800008386, 18344248255186535688600507495194211981022718803858650, 72198770808524759309871875344308930118869118982006008, 284207019536697729788560254220439744339643382607707710, 1118956798454956756627655423831115616024498160027663414, 4406196097592559931377055334700804678837764010080813780, 17353409506730383909378583467921257398893785398774976972, 68355718939914055041985367979947836886203482442149491320, 269297635353282005479319193908853937119069486421571119136, 1061100477246866235874623102099093001776756993436682196940
Generating function in Maple syntax:
1/2*(-(x-1)^3*(-4*x+1)^(1/2)+3*x^3-3*x^2+3*x-1)/(x^3-2*x^2+3*x-1)/x
Generating function in latex syntax:
\frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}+3 x^{3}-3 x^{2}+3 x -1}{2 \left(x^{3}-2 x^{2}+3 x -1\right) x}
Generating function in sympy syntax:
(3*x**3/2 - 3*x**2/2 + 3*x/2 - sqrt(1 - 4*x)*(x - 1)**3/2 - 1/2)/(x*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)^2*x*F(x)^2-(3*x^3-3*x^2+3*x-1)*(x^3-2*x^2+3*x-1)*F(x)+x^6-4*x^5+12*x^4-17*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right)^{2} x F \! \left(x \right)^{2}-\left(3 x^{3}-3 x^{2}+3 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = 2*(2*n+1)/(n+6)*a(n)-(13*n+24)/(n+6)*a(1+n)+(52+23*n)/(n+6)*a(n+2)-(21*n+74)/(n+6)*a(n+3)+2*(4*n+19)/(n+6)*a(n+4), n >= 5
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(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +6}-\frac{\left(13 n +24\right) a \! \left(1+n \right)}{n +6}+\frac{\left(52+23 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(21 n +74\right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(4 n +19\right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22032/
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[8,x]
F[4,x] = F[23,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[11,x]
F[19,x] = F[14,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[20,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[25,x]*F[4,x]*F[8,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]^2*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{23}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25} \left(x \right)^{2} F_{8}\! \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_8(x))
Eq(F_4(x), F_23(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_11(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_20(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_25(x)*F_4(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)**2*F_8(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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"}}, {"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, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[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": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", 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