0123_0132_0213
Counting sequence:
1, 1, 2, 6, 21, 80, 322, 1346, 5783, 25372, 113174, 511649, 2338988, 10793251, 50205607, 235156609, 1108120540, 5249646137, 24987770893, 119443412277, 573125649031, 2759515312908, 13328311926552, 64559295743113, 313530998739472, 1526333617345412, 7447070497787110, 36409703715788374, 178353171835771153, 875224495042876048, 4302111437028045585, 21179758960764209606, 104422835762346295058, 515545296244964921875, 2548591216255899312655, 12614314530507829955718, 62506907518289214409666, 310074734585314878739226, 1539763425357422177017261, 7653639602018290171623707, 38079150678448982914730617, 189623484533399493029424701, 945070919841222048568819683, 4713973453579355720335599699, 23531224052397469497300240249, 117549776118304689917057455186, 587631676724178290062569994550, 2939556259867217233329188505431, 14714293799979510887244995880762, 73699876594824088009119834030749, 369362744911090178869134080161513, 1852201940247004540959372877204385, 9293151183810849679394343819865229, 46651770530091458612902461091103771, 234312389852105108839163887513768459, 1177433434684440909463589118615369674, 5919484413149671520314542197659920500, 29773550114238746121794191537936100623, 149819992443528293964668445490563215656, 754214466743374989015048793127769969526, 3798391594053013999050729352254830974734, 19137207072912974815687385120878737102465, 96455223523745244347723421136255745646694, 486335553753858794947622822379999566660185, 2453038162927885796387431720234420449461459, 12377296139702522418922691173702538852427704, 62473501184540310089501138244760103281799538, 315435155842204148006045458570091956759093670, 1593178073853652513769420392578603654277542115, 8049232979343488625528388377704047655818913813, 40679609123866944963592832134995170386689232832, 205649400620309429507825267839514938485597929039, 1039927321514644789729505397852271083601835743733, 5260172626492506909330522384584874262273604429594, 26614310248357527964782455428852272485775706752541, 134693151477840671461429815681243685502176218527177, 681848498310052171818052658961603963863665072539257, 3452545790959765632888393851739057288050695659552859, 17486279348042469583273662687352872061283503755303108, 88584779467159871606210829896103042974515163128496506, 448871427271620894403980263881806601764273819550236184, 2275010951384170508264773455941431792019068208165130329, 11532973898779255995467203518336708132810730281953225475, 58478080706035122116512645438335765014749961970814632484, 296576457335360004202363935460238290699336118122130803538, 1504422616179145012694046487732015181749183958046203899684, 7632917740644088986847931032386897224472716797605309367001, 38734402380202962335508669908623810265348181444540351616069, 196601467767334790985001214951402968655224612849451661064313, 998064029875779774945395249303578004548763377639891748960565, 5067689525344811260846209794740458448662172505677459446116707, 25735925931420489898740321615631193854679147691897816608112565, 130721225294378659504854854428263732381878735738212508994989183, 664090551532827051123499397968064784179612096345501036405872076, 3374285216602293545219410074617980696608713024065880825501425938, 17147785441650524733688446778453568075967551980891775574174142245, 87157461131243233445788645857977722028768384614948018824674043382, 443067545939894826941184482367894781817915673305854510145055965342, 2252696849185579617195693337059928542515976202866528532758853055156, 11455173007816282669754114891970177552063592056306515225608510624256, 58259309072912937175843767048035019379079087095879822625531537951149
Implicit equation for the generating function in Maple syntax:
(2*x^2+8*x-1)*F(x)^4+(x-5)*(x^2+9*x-1)*F(x)^3+(3*x^3-21*x^2+94*x-9)*F(x)^2+(x^3+12*x^2-82*x+7)*F(x)+3*x^2+26*x-2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}+8 x -1\right) F \! \left(x \right)^{4}+\left(x -5\right) \left(x^{2}+9 x -1\right) F \! \left(x \right)^{3}+\left(3 x^{3}-21 x^{2}+94 x -9\right) F \! \left(x \right)^{2}+\left(x^{3}+12 x^{2}-82 x +7\right) F \! \left(x \right)+3 x^{2}+26 x -2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 80
a(6) = 322
a(7) = 1346
a(8) = 5783
a(9) = 25372
a(10) = 113174
a(11) = 511649
a(12) = 2338988
a(13) = 10793251
a(14) = 50205607
a(15) = 235156609
a(16) = 1108120540
a(n+17) = 64*(n-1)*n*(n+1)/(n+15)/(n+17)/(n+16)*a(n)-8*n*(163*n+281)*(n+1)/(n+15)/(n+17)/(n+16)*a(n+1)+16*(n+1)*(37*n^2+596*n+6)/(n+15)/(n+17)/(n+16)*a(n+2)+12*(5473*n^3+38232*n^2+101151*n+98960)/(n+15)/(n+17)/(n+16)*a(n+3)+12*(8421*n^3+73066*n^2+197441*n+156436)/(n+15)/(n+17)/(n+16)*a(n+4)-6*(30245*n^3+368164*n^2+1510787*n+2103516)/(n+15)/(n+17)/(n+16)*a(n+5)+12*(7030*n^3+65845*n^2+121061*n-160182)/(n+15)/(n+17)/(n+16)*a(n+6)-3*(18065*n^3+226928*n^2+833127*n+778152)/(n+15)/(n+17)/(n+16)*a(n+7)+3*(1549*n^3+24230*n^2+180289*n+636096)/(n+15)/(n+17)/(n+16)*a(n+8)-(23413*n^3+652944*n^2+6074141*n+18810402)/(n+15)/(n+17)/(n+16)*a(n+9)+2*(20173*n^3+619098*n^2+6299948*n+21273837)/(n+15)/(n+17)/(n+16)*a(n+10)-(23947*n^3+758244*n^2+7955477*n+27658776)/(n+15)/(n+17)/(n+16)*a(n+11)+(12937*n^3+436518*n^2+4887083*n+18148686)/(n+15)/(n+17)/(n+16)*a(n+12)-(2447*n^3+84444*n^2+958105*n+3561192)/(n+15)/(n+17)/(n+16)*a(n+13)-2*(19*n^3+1200*n^2+21932*n+124167)/(n+15)/(n+17)/(n+16)*a(n+14)-(n^2+237*n+3302)/(n+17)/(n+16)*a(n+15)+(11*n+193)/(n+17)*a(n+16), n >= 17
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) = 80
a \! \left(6\right) = 322
a \! \left(7\right) = 1346
a \! \left(8\right) = 5783
a \! \left(9\right) = 25372
a \! \left(10\right) = 113174
a \! \left(11\right) = 511649
a \! \left(12\right) = 2338988
a \! \left(13\right) = 10793251
a \! \left(14\right) = 50205607
a \! \left(15\right) = 235156609
a \! \left(16\right) = 1108120540
a \! \left(n +17\right) = \frac{64 \left(n -1\right) n \left(n +1\right) a \! \left(n \right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{8 n \left(163 n +281\right) \left(n +1\right) a \! \left(n +1\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{16 \left(n +1\right) \left(37 n^{2}+596 n +6\right) a \! \left(n +2\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(5473 n^{3}+38232 n^{2}+101151 n +98960\right) a \! \left(n +3\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(8421 n^{3}+73066 n^{2}+197441 n +156436\right) a \! \left(n +4\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{6 \left(30245 n^{3}+368164 n^{2}+1510787 n +2103516\right) a \! \left(n +5\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(7030 n^{3}+65845 n^{2}+121061 n -160182\right) a \! \left(n +6\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{3 \left(18065 n^{3}+226928 n^{2}+833127 n +778152\right) a \! \left(n +7\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{3 \left(1549 n^{3}+24230 n^{2}+180289 n +636096\right) a \! \left(n +8\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(23413 n^{3}+652944 n^{2}+6074141 n +18810402\right) a \! \left(n +9\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{2 \left(20173 n^{3}+619098 n^{2}+6299948 n +21273837\right) a \! \left(n +10\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(23947 n^{3}+758244 n^{2}+7955477 n +27658776\right) a \! \left(n +11\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{\left(12937 n^{3}+436518 n^{2}+4887083 n +18148686\right) a \! \left(n +12\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(2447 n^{3}+84444 n^{2}+958105 n +3561192\right) a \! \left(n +13\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{2 \left(19 n^{3}+1200 n^{2}+21932 n +124167\right) a \! \left(n +14\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(n^{2}+237 n +3302\right) a \! \left(n +15\right)}{\left(n +17\right) \left(n +16\right)}+\frac{\left(11 n +193\right) a \! \left(n +16\right)}{n +17}, \quad n \geq 17
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/225/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[15,x]*F[3,x]
F[3,x] = F[1,x]+F[16,x]+F[4,x]
F[4,x] = F[15,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[4,x]+F[7,x,k[0]]+F[9,x,k[0]]
F[7,x,k[0]] = F[6,x,k[0]]*F[8,x,k[0]]
F[8,x,k[0]] = k[0]*x
F[9,x,k[0]] = F[10,x,k[0]]*F[15,x]
F[10,x,k[0]] = (F[11,x,k[0]]*k[0]-F[11,x,1])/(-1+k[0])
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[13,x,k[0]]+F[9,x,k[0]]
F[12,x,k[0]] = F[11,x,k[0]]*F[8,x,k[0]]
F[13,x,k[0]] = F[14,x,k[0]]*F[15,x]
F[14,x,k[0]] = (F[6,x,k[0]]*k[0]-F[6,x,1])/(-1+k[0])
F[15,x] = x
F[16,x] = F[15,x]*F[17,x]
F[17,x] = F[11,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_{15}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{15}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = y x
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{15}\! \left(x \right)
F_{10}\! \left(x , y\right) = -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{15}\! \left(x \right)
F_{14}\! \left(x , y\right) = -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \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_15(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_16(x) + F_4(x))
Eq(F_4(x), F_15(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_4(x) + F_7(x, k_0) + F_9(x, k_0))
Eq(F_7(x, k_0), F_6(x, k_0)*F_8(x, k_0))
Eq(F_8(x, k_0), k_0*x)
Eq(F_9(x, k_0), F_10(x, k_0)*F_15(x))
Eq(F_10(x, k_0), (-k_0*F_11(x, k_0) + F_11(x, 1))/(1 - k_0))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0) + F_13(x, k_0) + F_9(x, k_0))
Eq(F_12(x, k_0), F_11(x, k_0)*F_8(x, k_0))
Eq(F_13(x, k_0), F_14(x, k_0)*F_15(x))
Eq(F_14(x, k_0), (-k_0*F_6(x, k_0) + F_6(x, 1))/(1 - k_0))
Eq(F_15(x), x)
Eq(F_16(x), F_15(x)*F_17(x))
Eq(F_17(x), F_11(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 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]]}], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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, 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