0213_0231_0312
Counting sequence:
1, 1, 2, 6, 21, 79, 310, 1251, 5151, 21536, 91137, 389510, 1678565, 7284975, 31811311, 139661231, 616097345, 2729518734, 12139620771, 54181709667, 242603277963, 1089487427660, 4906014417063, 22147650846291, 100216641145675, 454459064260660, 2065042593916742, 9401243324422881, 42875744357878842, 195866829629921243, 896168786052340513, 4106378057645706969, 18842216287469029209, 86571632542075561372, 398252971818839729701, 1834230031710111704233, 8457336309060333783177, 39036781751389787223870, 180365025909020114710318, 834156339529680276717847, 3861354829006518592202775, 17890016475767651524211318, 82955156182847110029391309, 384965308565262186036340741, 1787846920003150484964427703, 8309128774814217256594245696, 38644156102086331768145299556, 179846874475675644868684719419, 837531160480072714376285471173, 3902716710462484596977339900742, 18196616727608809539718967388116, 84891070023465093054011674929698, 396252325798051432595523729485340, 1850594713750955192982517073668076, 8647133724163890140115080795308801, 40424673740532914469613381396895880, 189071859222296553057698973868318356, 884720727574834524846856239509450608, 4141691188878143783837951626626597144, 19397020791940170510776031380569840111, 90880784266421477975044518154476753496, 425973931901781545451940501306479569323, 1997387733206820048729746707881668129871, 9369250876917388274411663089225256177013, 43964837047727968079984769881370085165317, 206376090421184024427002071455430194905412, 969085341568536369929425944689226461523325, 4552070890569644800382561100030529857331503, 21389280030796086049430585496486883475556357, 100535504247806731415207493746197048825025264, 472688521238110709903982009834722497663300256, 2223101385887048675499716309948923387279715605, 10458479956280682005756840919678898543323689901, 49215235076962624604227572203147236681731511372, 231658906530356443915445868691507831502814008584, 1090721208363775336039562062251991757174760949898, 5136778362544282932548639886227128607045891832384, 24197880264794537363559740838030277282270257823474, 114017225135275946888186832149802778161300418840390, 537362783787320614241047500832675145710620095102499, 2533180301945978033246499751954032894102090430904667, 11944377981352221303828355533155637016851143519238098, 56332302443148553535425394956714299475546706247037377, 265733121628425401968083448441793823699507199123664951, 1253793100306148685645698427067777146909595569631689611, 5916922965212061851411610687429920458386577803719745534, 27928893884914032587515694053041503585929146989234278744, 131855222859059661357537896954919048972821934302323946447, 622622462909988910334666320788585947828950363306955678915, 2940588204592176849304860044043134042151079731840024048868, 13890689617148503697030354182781585666390490698256521726592, 65628401472544653169731254424058750707089478723212594357604, 310124856225476893359611187010858473447422625990938897329660, 1465738242169029600479956196290423051531967546302590220945220, 6928668287313512366792790284953155781435806821417290311122178, 32757827542332980092176920531588710430089449188492731918324629, 154899814526137829141758400485537436070260711255868494443932533, 732581156882844773982593113328401137748184687408614860694622588, 3465199405891063029970991385368011801610255665387984963587168526, 16393323283387799660691525930127439965283344620947030297502141468, 77565883533923975381211004729200332643592833424364684410083412402
Implicit equation for the generating function in Maple syntax:
x*F(x)^4+(-x-1)*F(x)^3+(x^2-2*x+3)*F(x)^2+(2*x-3)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
x F \! \left(x \right)^{4}+\left(-x -1\right) F \! \left(x \right)^{3}+\left(x^{2}-2 x +3\right) F \! \left(x \right)^{2}+\left(2 x -3\right) F \! \left(x \right)+1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 79
a(6) = 310
a(7) = 1251
a(8) = 5151
a(n+9) = n*(2*n+3)*(2*n-1)/(n+10)/(n+8)/(2*n+17)*a(n)+(8*n^3+83*n^2+145*n+55)/(n+10)/(n+8)/(2*n+17)*a(n+1)+1/12*(1151*n^3+8520*n^2+22579*n+18546)/(n+10)/(n+8)/(2*n+17)*a(n+2)-1/24*(2233*n^3+55122*n^2+279635*n+395634)/(n+10)/(n+8)/(2*n+17)*a(n+3)+1/24*(62879*n^3+790140*n^2+3299443*n+4570050)/(n+10)/(n+8)/(2*n+17)*a(n+4)-1/24*(67265*n^3+1013004*n^2+5056489*n+8372154)/(n+10)/(n+8)/(2*n+17)*a(n+5)+1/6*(7612*n^3+136431*n^2+810569*n+1597989)/(n+10)/(n+8)/(2*n+17)*a(n+6)-1/12*(3640*n^3+76029*n^2+526883*n+1212402)/(n+10)/(n+8)/(2*n+17)*a(n+7)+1/12*(458*n^3+10893*n^2+85993*n+225450)/(n+10)/(n+8)/(2*n+17)*a(n+8), 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) = 79
a \! \left(6\right) = 310
a \! \left(7\right) = 1251
a \! \left(8\right) = 5151
a \! \left(n +9\right) = \frac{n \left(2 n +3\right) \left(2 n -1\right) a \! \left(n \right)}{\left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(8 n^{3}+83 n^{2}+145 n +55\right) a \! \left(n +1\right)}{\left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(1151 n^{3}+8520 n^{2}+22579 n +18546\right) a \! \left(n +2\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(2233 n^{3}+55122 n^{2}+279635 n +395634\right) a \! \left(n +3\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(62879 n^{3}+790140 n^{2}+3299443 n +4570050\right) a \! \left(n +4\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(67265 n^{3}+1013004 n^{2}+5056489 n +8372154\right) a \! \left(n +5\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(7612 n^{3}+136431 n^{2}+810569 n +1597989\right) a \! \left(n +6\right)}{6 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(3640 n^{3}+76029 n^{2}+526883 n +1212402\right) a \! \left(n +7\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(458 n^{3}+10893 n^{2}+85993 n +225450\right) a \! \left(n +8\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}, \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_placements_tracked_fusion
Tree: http://permpal.com/tree/304/
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[9,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[10,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]^2*F[9,x]
F[8,x] = F[1,x]+F[6,x]
F[9,x] = x
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]*F[9,x]
F[12,x] = F[13,x]+F[39,x]
F[13,x] = F[14,x]+F[19,x]
F[14,x] = F[15,x,1]
F[15,x,k[0]] = F[16,x,k[0]]
F[16,x,k[0]] = F[17,x,k[0]]*F[18,x,k[0]]
F[17,x,k[0]] = k[0]*x
F[18,x,k[0]] = F[1,x]+F[15,x,k[0]]
F[19,x] = F[20,x,1]
F[20,x,k[0]] = F[2,x]+F[21,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[17,x,k[0]]*F[23,x,k[0]]
F[23,x,k[0]] = F[20,x,k[0]]+F[24,x,k[0]]
F[24,x,k[0]] = F[25,x,k[0]]+F[26,x,k[0]]
F[25,x,k[0]] = F[18,x,k[0]]*F[6,x]
F[26,x,k[0]] = F[10,x]+F[27,x,k[0]]
F[27,x,k[0]] = F[28,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]*F[8,x]*F[9,x]
F[29,x,k[0]] = F[30,x,k[0]]+F[35,x,k[0]]
F[30,x,k[0]] = F[31,x,k[0]]+F[34,x,k[0]]
F[31,x,k[0]] = F[32,x,k[0]]
F[32,x,k[0]] = F[17,x,k[0]]*F[33,x,k[0]]
F[33,x,k[0]] = F[14,x]+F[31,x,k[0]]
F[34,x,k[0]] = -k[0]*(F[21,x,1]-F[21,x,k[0]])/(-1+k[0])
F[35,x,k[0]] = F[36,x,k[0]]+F[38,x,k[0]]
F[36,x,k[0]] = F[37,x,k[0]]
F[37,x,k[0]] = F[14,x]*F[15,x,k[0]]*F[6,x]
F[38,x,k[0]] = -k[0]*(F[27,x,1]-F[27,x,k[0]])/(-1+k[0])
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[14,x]*F[6,x]
F[41,x] = F[26,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_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{10}\! \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)^{2} F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x , 1\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = y x
F_{18}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)
F_{19}\! \left(x \right) = F_{20}\! \left(x , 1\right)
F_{20}\! \left(x , y\right) = F_{2}\! \left(x \right)+F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{20}\! \left(x , y\right)+F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{25}\! \left(x , y\right)+F_{26}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{6}\! \left(x \right)
F_{26}\! \left(x , y\right) = F_{10}\! \left(x \right)+F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{17}\! \left(x , y\right) F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{14}\! \left(x \right)+F_{31}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = -\frac{y \left(F_{21}\! \left(x , 1\right)-F_{21}\! \left(x , y\right)\right)}{-1+y}
F_{35}\! \left(x , y\right) = F_{36}\! \left(x , y\right)+F_{38}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{15}\! \left(x , y\right) F_{6}\! \left(x \right)
F_{38}\! \left(x , y\right) = -\frac{y \left(F_{27}\! \left(x , 1\right)-F_{27}\! \left(x , y\right)\right)}{-1+y}
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right) F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{26}\! \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))
Eq(F_3(x), F_4(x)*F_9(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_10(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)**2*F_9(x))
Eq(F_8(x), F_1(x) + F_6(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x)*F_9(x))
Eq(F_12(x), F_13(x) + F_39(x))
Eq(F_13(x), F_14(x) + F_19(x))
Eq(F_14(x), F_15(x, 1))
Eq(F_15(x, k_0), F_16(x, k_0))
Eq(F_16(x, k_0), F_17(x, k_0)*F_18(x, k_0))
Eq(F_17(x, k_0), k_0*x)
Eq(F_18(x, k_0), F_1(x) + F_15(x, k_0))
Eq(F_19(x), F_20(x, 1))
Eq(F_20(x, k_0), F_2(x) + F_21(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_17(x, k_0)*F_23(x, k_0))
Eq(F_23(x, k_0), F_20(x, k_0) + F_24(x, k_0))
Eq(F_24(x, k_0), F_25(x, k_0) + F_26(x, k_0))
Eq(F_25(x, k_0), F_18(x, k_0)*F_6(x))
Eq(F_26(x, k_0), F_10(x) + F_27(x, k_0))
Eq(F_27(x, k_0), F_28(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0)*F_8(x)*F_9(x))
Eq(F_29(x, k_0), F_30(x, k_0) + F_35(x, k_0))
Eq(F_30(x, k_0), F_31(x, k_0) + F_34(x, k_0))
Eq(F_31(x, k_0), F_32(x, k_0))
Eq(F_32(x, k_0), F_17(x, k_0)*F_33(x, k_0))
Eq(F_33(x, k_0), F_14(x) + F_31(x, k_0))
Eq(F_34(x, k_0), -F_21(x, 1) + (k_0*F_21(x, k_0) - F_21(x, 1))/(k_0 - 1))
Eq(F_35(x, k_0), F_36(x, k_0) + F_38(x, k_0))
Eq(F_36(x, k_0), F_37(x, k_0))
Eq(F_37(x, k_0), F_14(x)*F_15(x, k_0)*F_6(x))
Eq(F_38(x, k_0), -F_27(x, 1) + (k_0*F_27(x, k_0) - F_27(x, 1))/(k_0 - 1))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_14(x)*F_6(x))
Eq(F_41(x), F_26(x, 1))
Pack JSON:
{"expansion_strats": [[{"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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_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:
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