0132_1032_1320
Counting sequence:
1, 1, 2, 6, 21, 77, 287, 1080, 4094, 15611, 59811, 230048, 887674, 3434510, 13319262, 51756304, 201467116, 785439987, 3066304355, 11985287488, 46898578706, 183697694922, 720180824586, 2825786740784, 11096029933044, 43601382335006, 171440847547406, 674511447238784, 2655260104721732, 10458061203086588, 41210408379154140, 162465433255662112, 640768669759885784, 2528231394933740867, 9979243071401655971, 39403467164745340800, 155639171630179312258, 614954913036988553842, 2430526671171195639570, 9609120904174779803760, 38000294613124626678820, 150315801569300834730666, 594745829033834994736026, 2353760964164559134985728, 9317360686541423366839564, 36890859884688560546807684, 146095203933032934594401124, 578682958219824196530210528, 2292606150193211401695405032, 9084455971493049077520540654, 36003604844092015263649172750, 142714368048370796117372864384, 565797842214530086946092383380, 2243489958515810472227535228964, 8897208131680002806769886360612, 35289686562333750774761726263776, 139992271296100968114380455708040, 555418597733368911429940479859420, 2203915161689553571244258000619836, 8746323209434188274160461463581440, 34714471396836121983240820162027464, 137799694049597131158639266343279992, 547062531959646124077264631946996280, 2172076641627975126713439826348321856, 8625043265308791536884843460312298416, 34252632604661501013919579756097596259, 136041623062425917888709778858207305763, 540372854621524299399014629126091825920, 2146633329871460039733855908319309128802, 8528323036416531585971719708511356925954, 33885172615256390568221853235856847530338, 134646463630264249671990986147556389310704, 535079530716739227767049895082259072952836, 2126565922070691394968574154929891831145586, 8452311986625320780490602052396577189706594, 33597534780259575865460653129923901177048320, 133559145856708485000862444025709572054640572, 530974077525814316349221654443502620094624948, 2111084613971809866204814624332237278725963412, 8394016104004066653027080716102723240628043616, 33378362098954955649975740195853429348608189512, 132736563356550366404333346197107394441927737722, 527892783444613957871759945465918090269372244250, 2099567292093177704989979842933474385909368022912, 8351069987836193461580877149566274865955918717340, 33218655913469988060441793069534599529389429771884, 132144464451116115646030052588339967971682453119212, 525705208551821817013722568182448479433563292384160, 2091516955263079505970716203461171421121517396874520, 8321579043052322358014970985950203499595571368275428, 33111190496085425872452055527488914144920012756670468, 131755278473565567845804596313494861343131719867335424, 524306106287405066689809267534028883436563106891027512, 2086531666321512191992710449845999536404755798873681352, 8304007617566962013100538985520378527392686148562210312, 33050096268440227484910890336781053557292677568410899904, 131546561670174171304353994058165544656903482069619933136, 523609623969506545154571178323815581487544840542930869582, 2084282893464665401734288464772544139362446572538280957070, 8297098055930681230798856302413747741534779749079119921408, 33030557055188124176438932828529360839578889525088495310516
Generating function in Maple syntax:
((-4*x^4+10*x^3-7*x^2+x)*(1-4*x)^(1/2)+24*x^4-58*x^3+57*x^2-19*x+2)/(32*x^4-72*x^3+64*x^2-20*x+2)
Generating function in latex syntax:
\frac{\left(-4 x^{4}+10 x^{3}-7 x^{2}+x \right) \sqrt{1-4 x}+24 x^{4}-58 x^{3}+57 x^{2}-19 x +2}{32 x^{4}-72 x^{3}+64 x^{2}-20 x +2}
Generating function in sympy syntax:
(24*x**4 - 58*x**3 + 57*x**2 - 19*x + sqrt(1 - 4*x)*(-4*x**4 + 10*x**3 - 7*x**2 + x) + 2)/(32*x**4 - 72*x**3 + 64*x**2 - 20*x + 2)
Implicit equation for the generating function in Maple syntax:
(4*x-1)*(4*x^3-8*x^2+6*x-1)*F(x)^2-(4*x-1)*(6*x^3-13*x^2+11*x-2)*F(x)+x^5+6*x^4-21*x^3+25*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x -1\right) \left(4 x^{3}-8 x^{2}+6 x -1\right) F \! \left(x \right)^{2}-\left(4 x -1\right) \left(6 x^{3}-13 x^{2}+11 x -2\right) F \! \left(x \right)+x^{5}+6 x^{4}-21 x^{3}+25 x^{2}-9 x +1 = 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) = 287
a(7) = 1080
a(n+7) = 32*(2*n-1)/(n+6)*a(n)-16*(19*n+11)/(n+6)*a(n+1)+8*(75*n+113)/(n+6)*a(n+2)-4*(157*n+374)/(n+6)*a(n+3)+4*(91*n+298)/(n+6)*a(n+4)-2*(56*n+233)/(n+6)*a(n+5)+(86+17*n)/(n+6)*a(n+6), n >= 8
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) = 287
a \! \left(7\right) = 1080
a \! \left(n +7\right) = \frac{32 \left(2 n -1\right) a \! \left(n \right)}{n +6}-\frac{16 \left(19 n +11\right) a \! \left(n +1\right)}{n +6}+\frac{8 \left(75 n +113\right) a \! \left(n +2\right)}{n +6}-\frac{4 \left(157 n +374\right) a \! \left(n +3\right)}{n +6}+\frac{4 \left(91 n +298\right) a \! \left(n +4\right)}{n +6}-\frac{2 \left(56 n +233\right) a \! \left(n +5\right)}{n +6}+\frac{\left(86+17 n \right) a \! \left(n +6\right)}{n +6}, \quad n \geq 8
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/284/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[19,x]+F[4,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[3,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[16,x]*F[3,x]*F[8,x]
F[8,x] = F[1,x]+F[15,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[14,x]*F[8,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[8,x]
F[14,x] = x
F[15,x] = F[14,x]*F[8,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[22,x]+F[3,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[11,x]*F[14,x]*F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[11,x]*F[14,x]*F[23,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{3}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{23}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_19(x) + F_4(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_3(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_16(x)*F_3(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_15(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_14(x)*F_8(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_8(x))
Eq(F_14(x), x)
Eq(F_15(x), F_14(x)*F_8(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_14(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_3(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_11(x)*F_14(x)*F_23(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_11(x)*F_14(x)*F_23(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"}, {"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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_and_row_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, 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], "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, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 1], [0, 1], [2, 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, 1, 3, 2], "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_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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 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, 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, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"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], "pos": [[3, 1]]}, {"patt": [0, 1], "pos": [[0, 0], [2, 0]]}, {"patt": [0, 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