01342_01423_01432_02341_02413_02431_03412_03421_10342_10423_10432_12403_13402_20413_21403
Counting sequence:
1, 1, 2, 6, 24, 105, 478, 2237, 10707, 52202, 258410, 1295456, 6563870, 33561319, 172948169, 897327948, 4683652331, 24576380580, 129568975877, 685997450424, 3645876288227, 19443934362723, 104024291432744, 558131618685401, 3002527371885829, 16191802756790142, 87514700746285932, 473994056082552088, 2572212922144935412, 13983814855713388993, 76151315565420353141, 415351101863480360491, 2268804572341745915302, 12410347672491685611997, 67973665798055553618768, 372764939688073106575602, 2046621106853851359544142, 11249174229601868597037776, 61895567832107480399733587, 340903136186229537800267644, 1879375346198223193664564997, 10370202620767282435881897317, 57270854151106450392510944918, 316545084630032360159817127732, 1750962889293001659796872306662, 9692655619505470936599979847841, 53693175165276705835322854988171, 297640984670808143320425833566538, 1651016495059376227501243884132463, 9163968060235624563920490774903866, 50895375924051637943890953002585997, 282829989043046153067470507148123100, 1572589275830386124879837513393372095, 8748604435787017012846490010473367207, 48695324819120469534422678501468355990, 271176863799067762556520246443270795085, 1510870279173234340330190656363455901747, 8421774353237209866921072554340175700560, 46965070526754595079163606623553388921348, 262020178318216189746295509535535993511812, 1462435199144675335380443407326035973057796, 8165724412197454149896420083445148204567677, 45612444130174553164735177297905214460295298, 254880799993274161379326484882328910679351590, 1424790183729313429167654721550059211354946912, 7967458733023738444764468597543078629575260773, 44569644343022610336330185715335855871170005073, 249404612934065205924572011544424675395021080686, 1396083805030454078192662696399579400742788470561, 7817287814899801394874816532546717853701721643772, 43785912629917141288650154946945053474285872278682, 245325477111975001454141163928409836084507566964226, 1374919419605041860028782218657862513211412711922760, 7707876384723930855758272705287367072220915460038164, 43222693707700733418290559131454636878805597533578482, 242440591522161256360750961673406339661740406519706280, 1360229554626814993338511101423324433174671736271194130, 7633601985569720633479505832766657758007133963409095662, 42850355630533873125107418626363058750465929438068816335, 240593698584093967924547227642897666584276420591613800756, 1351189785318602984564889932679517871789912347746615082145, 7590112786490881083585838157122231458068003996127018220629, 42645916500243461399526778760889200899147608209553650947210, 239663382751114570052642092023371458081378359649536426525630, 1347158431620395495467343768342200540712922594828946168538256, 7574016446013377291342110475861732622667560953239194718281365, 42591437333835533045995027179824724225393075697944490013660792, 239554759972325166928237672031535762462580170242448123716943111, 1347633538274341690303160566924390892227223753052038112016502075, 7582657186763624073739465818895866285834903663194570139866112210, 42672865722255124184701025700926887309288029588282311720323352780, 240193473750686876671609447019514332082186192284681521299818656152, 1352221671660840981400521852650869546328032123069392212577823434316, 7613953479770988527912500988507568371167288373563518949584866765379, 42879190725722575358277212982584985252036791324422232148245564297521, 241521291245768596121818267557834533198137526194525601744435248765131, 1360614951468317803967611372240692101902823040469420701602379411488242, 7666278156300427033132735017400456603320544662668470394487522487205827, 43201816595731487866502852398609468058729591819097731640577030829375734, 243492829177855151731730963625213830502475141795763757646357188325331766, 1372573921431162131842371146465641251376581264135081079392941930720516260
Implicit equation for the generating function in Maple syntax:
x^2*(x^4+x^3-x^2-4*x+4)*F(x)^3+x*(x^3+7*x^2-11*x+2)*F(x)^2+(-3*x^3+6*x^2-1)*F(x)-x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x^{4}+x^{3}-x^{2}-4 x +4\right) F \! \left(x \right)^{3}+x \left(x^{3}+7 x^{2}-11 x +2\right) F \! \left(x \right)^{2}+\left(-3 x^{3}+6 x^{2}-1\right) F \! \left(x \right)-x^{2}-x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 105
a(6) = 478
a(7) = 2237
a(8) = 10707
a(9) = 52202
a(10) = 258410
a(11) = 1295456
a(12) = 6563870
a(13) = 33561319
a(14) = 172948169
a(15) = 897327948
a(16) = 4683652331
a(17) = 24576380580
a(18) = 129568975877
a(19) = 685997450424
a(20) = 3645876288227
a(21) = 19443934362723
a(n+22) = 9/160*(3*n+5)*(3*n+4)/(n+23)/(n+22)*a(n)-1/160*(369*n^2+2457*n+3688)/(n+23)/(n+22)*a(n+1)-3/160*(67*n^2+33*n-656)/(n+23)/(n+22)*a(n+2)-1/240*(2744*n^2+33471*n+98770)/(n+23)/(n+22)*a(n+3)+1/80*(929*n^2+9578*n+35720)/(n+23)/(n+22)*a(n+4)+1/240*(1447*n^2+36129*n+102314)/(n+23)/(n+22)*a(n+5)+1/240*(5903*n^2+15600*n-139652)/(n+23)/(n+22)*a(n+6)-1/240*(13484*n^2+83163*n-217742)/(n+23)/(n+22)*a(n+7)+1/480*(20909*n^2+244371*n+243100)/(n+23)/(n+22)*a(n+8)+1/480*(9667*n^2+67449*n+93164)/(n+23)/(n+22)*a(n+9)-1/160*(35247*n^2+552635*n+1943900)/(n+23)/(n+22)*a(n+10)+1/80*(45192*n^2+804721*n+3237910)/(n+23)/(n+22)*a(n+11)-1/240*(228277*n^2+4673457*n+22605932)/(n+23)/(n+22)*a(n+12)+1/240*(245975*n^2+5646738*n+31324906)/(n+23)/(n+22)*a(n+13)-1/240*(110825*n^2+2599392*n+14351614)/(n+23)/(n+22)*a(n+14)-1/120*(44510*n^2+1464093*n+12014848)/(n+23)/(n+22)*a(n+15)+1/480*(380171*n^2+12316059*n+99737140)/(n+23)/(n+22)*a(n+16)-1/160*(97265*n^2+3273035*n+27474928)/(n+23)/(n+22)*a(n+17)+1/160*(31679*n^2+1079397*n+9104220)/(n+23)/(n+22)*a(n+18)+1/120*(3178*n^2+151065*n+1745060)/(n+23)/(n+22)*a(n+19)-1/40*(1677*n^2+70659*n+744136)/(n+23)/(n+22)*a(n+20)+1/30*(349*n+7466)/(n+23)*a(n+21), n >= 22
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) = 24
a \! \left(5\right) = 105
a \! \left(6\right) = 478
a \! \left(7\right) = 2237
a \! \left(8\right) = 10707
a \! \left(9\right) = 52202
a \! \left(10\right) = 258410
a \! \left(11\right) = 1295456
a \! \left(12\right) = 6563870
a \! \left(13\right) = 33561319
a \! \left(14\right) = 172948169
a \! \left(15\right) = 897327948
a \! \left(16\right) = 4683652331
a \! \left(17\right) = 24576380580
a \! \left(18\right) = 129568975877
a \! \left(19\right) = 685997450424
a \! \left(20\right) = 3645876288227
a \! \left(21\right) = 19443934362723
a \! \left(n +22\right) = \frac{9 \left(3 n +5\right) \left(3 n +4\right) a \! \left(n \right)}{160 \left(n +23\right) \left(n +22\right)}-\frac{\left(369 n^{2}+2457 n +3688\right) a \! \left(n +1\right)}{160 \left(n +23\right) \left(n +22\right)}-\frac{3 \left(67 n^{2}+33 n -656\right) a \! \left(n +2\right)}{160 \left(n +23\right) \left(n +22\right)}-\frac{\left(2744 n^{2}+33471 n +98770\right) a \! \left(n +3\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(929 n^{2}+9578 n +35720\right) a \! \left(n +4\right)}{80 \left(n +23\right) \left(n +22\right)}+\frac{\left(1447 n^{2}+36129 n +102314\right) a \! \left(n +5\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(5903 n^{2}+15600 n -139652\right) a \! \left(n +6\right)}{240 \left(n +23\right) \left(n +22\right)}-\frac{\left(13484 n^{2}+83163 n -217742\right) a \! \left(n +7\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(20909 n^{2}+244371 n +243100\right) a \! \left(n +8\right)}{480 \left(n +23\right) \left(n +22\right)}+\frac{\left(9667 n^{2}+67449 n +93164\right) a \! \left(n +9\right)}{480 \left(n +23\right) \left(n +22\right)}-\frac{\left(35247 n^{2}+552635 n +1943900\right) a \! \left(n +10\right)}{160 \left(n +23\right) \left(n +22\right)}+\frac{\left(45192 n^{2}+804721 n +3237910\right) a \! \left(n +11\right)}{80 \left(n +23\right) \left(n +22\right)}-\frac{\left(228277 n^{2}+4673457 n +22605932\right) a \! \left(n +12\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(245975 n^{2}+5646738 n +31324906\right) a \! \left(n +13\right)}{240 \left(n +23\right) \left(n +22\right)}-\frac{\left(110825 n^{2}+2599392 n +14351614\right) a \! \left(n +14\right)}{240 \left(n +23\right) \left(n +22\right)}-\frac{\left(44510 n^{2}+1464093 n +12014848\right) a \! \left(n +15\right)}{120 \left(n +23\right) \left(n +22\right)}+\frac{\left(380171 n^{2}+12316059 n +99737140\right) a \! \left(n +16\right)}{480 \left(n +23\right) \left(n +22\right)}-\frac{\left(97265 n^{2}+3273035 n +27474928\right) a \! \left(n +17\right)}{160 \left(n +23\right) \left(n +22\right)}+\frac{\left(31679 n^{2}+1079397 n +9104220\right) a \! \left(n +18\right)}{160 \left(n +23\right) \left(n +22\right)}+\frac{\left(3178 n^{2}+151065 n +1745060\right) a \! \left(n +19\right)}{120 \left(n +23\right) \left(n +22\right)}-\frac{\left(1677 n^{2}+70659 n +744136\right) a \! \left(n +20\right)}{40 \left(n +23\right) \left(n +22\right)}+\frac{\left(349 n +7466\right) a \! \left(n +21\right)}{30 n +690}, \quad n \geq 22
Specification 1
Strategy pack name: col_placements_tracked_fusion
Tree: https://permpal.com/tree/24304/
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[4,x]
F[3,x] = x
F[4,x] = F[1,x]+F[25,x]+F[5,x]
F[5,x] = F[3,x]*F[6,x]
F[6,x] = F[7,x,1]
F[7,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[23,x,k[0]]+F[8,x,k[0]]
F[8,x,k[0]] = F[3,x]*F[9,x,k[0]]
F[9,x,k[0]] = 1/(-1+k[0])*(F[7,x,k[0]]*k[0]-F[7,x,1])
F[10,x,k[0]] = F[11,x,k[0]]*F[3,x]
F[11,x,k[0]] = 1/(-1+k[0])*(F[12,x,k[0]]*k[0]-F[12,x,1])
F[12,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[13,x,k[0]]+F[8,x,k[0]]
F[13,x,k[0]] = F[14,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]*F[16,x,k[0]]*F[18,x,k[0]]
F[15,x,k[0]] = k[0]*x
F[16,x,k[0]] = F[15,x,k[0]]+F[17,x]
F[17,x] = F[1,x]+F[3,x]
F[18,x,k[0]] = F[1,x]+F[19,x,k[0]]+F[20,x,k[0]]
F[19,x,k[0]] = F[3,x]*F[7,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]
F[21,x,k[0]] = F[15,x,k[0]]*F[18,x,k[0]]*F[22,x,k[0]]
F[22,x,k[0]] = F[1,x]+F[15,x,k[0]]
F[23,x,k[0]] = F[24,x,k[0]]
F[24,x,k[0]] = F[12,x,k[0]]*F[15,x,k[0]]*F[22,x,k[0]]
F[25,x] = F[26,x]*F[3,x]
F[26,x] = F[12,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_{3}\! \left(x \right) F_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \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 , 1\right)
F_{7}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{-1+y}
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{11}\! \left(x , y\right) = \frac{F_{12}\! \left(x , y\right) y -F_{12}\! \left(x , 1\right)}{-1+y}
F_{12}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{18}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = y x
F_{16}\! \left(x , y\right) = F_{15}\! \left(x , y\right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{3}\! \left(x \right)
F_{18}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \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_3(x)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_1(x) + F_25(x) + F_5(x))
Eq(F_5(x), F_3(x)*F_6(x))
Eq(F_6(x), F_7(x, 1))
Eq(F_7(x, k_0), F_1(x) + F_10(x, k_0) + F_23(x, k_0) + F_8(x, k_0))
Eq(F_8(x, k_0), F_3(x)*F_9(x, k_0))
Eq(F_9(x, k_0), (k_0*F_7(x, k_0) - F_7(x, 1))/(k_0 - 1))
Eq(F_10(x, k_0), F_11(x, k_0)*F_3(x))
Eq(F_11(x, k_0), (k_0*F_12(x, k_0) - F_12(x, 1))/(k_0 - 1))
Eq(F_12(x, k_0), F_1(x) + F_10(x, k_0) + F_13(x, k_0) + F_8(x, k_0))
Eq(F_13(x, k_0), F_14(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0)*F_16(x, k_0)*F_18(x, k_0))
Eq(F_15(x, k_0), k_0*x)
Eq(F_16(x, k_0), F_15(x, k_0) + F_17(x))
Eq(F_17(x), F_1(x) + F_3(x))
Eq(F_18(x, k_0), F_1(x) + F_19(x, k_0) + F_20(x, k_0))
Eq(F_19(x, k_0), F_3(x)*F_7(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0))
Eq(F_21(x, k_0), F_15(x, k_0)*F_18(x, k_0)*F_22(x, k_0))
Eq(F_22(x, k_0), F_1(x) + F_15(x, k_0))
Eq(F_23(x, k_0), F_24(x, k_0))
Eq(F_24(x, k_0), F_12(x, k_0)*F_15(x, k_0)*F_22(x, k_0))
Eq(F_25(x), F_26(x)*F_3(x))
Eq(F_26(x), F_12(x, 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": false, "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": "col_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 4, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], 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