0321_1032_02413_13024_30214_30241_31024
Counting sequence:
1, 1, 2, 6, 22, 84, 320, 1212, 4576, 17256, 65048, 245184, 924160, 3483408, 13129952, 49490592, 186544480, 703140672, 2650342784, 9989916864, 37654917376, 141932392320, 534984681344, 2016513669120, 7600829555200, 28649748728064, 107989278831104, 407043163037184, 1534264682280448, 5783092136297472, 21798164973203456, 82163656569928704, 309698842505322496, 1167345479171868672, 4400066389397559296, 16585136599698604032, 62514228579247243264, 235634403814882775040, 888174956695015841792, 3347791073496750759936, 12618803296919642791936, 47563958786720444203008, 179282466192106741858304, 675768029066953865871360, 2547167265200978441863168, 9601018097688972443615232, 36189044108525773438287872, 136407087265470786844753920, 514158191093700821347336192, 1938012538558651700266205184, 7304935844007726709192392704, 27534438825022048180576321536, 103785349741406932193131233280, 391197325262267471608178540544, 1474537087109667796801216839680, 5557961368483773657884303032320, 20949581291379568808942721040384, 78965096586097683055620596957184, 297642534813693052191602189533184, 1121901730769458895447063493869568, 4228775615996408949100082649628672, 15939491597166023053279474096275456, 60080603807648802201358399044583424, 226461360570211036369956149202518016, 853599074927774334525829749346926592, 3217464466710428456663567398323879936, 12127564214406093612942843791689121792, 45712335068898923118176367262977490944, 172303155069595173319611170050917007360, 649460965015018370413560410425861865472, 2448008249807552027994308965623866064896, 9227258779112699503747916709790079778816, 34780235966690786965025773065056989216768, 131096877507863038920641325086184333901824, 494142458054946560896077720125276589916160, 1862567389051193697960207708280158135779328, 7020561018804874812324316998945288918925312, 26462546970646991259720291231621811695255552, 99745075970140361622839823649958929796956160, 375968352227201506013781931577207195342733312, 1417134635485687367412615100353384744258371584, 5341594746462951314074910228247990090904633344, 20134032237284043246406001955417336274900484096, 75891053771988870542162600052144425205906800640, 286055568737870498673365042128181129031863762944, 1078227068131038546097000263980146539781517475840, 4064153044039459513101326762911934499605908029440, 15318980995353594609678148190439228374011455995904, 57741717940513203803637623097396904486698760536064, 217645415953780033803652217257460750177713451433984, 820369202289669732443605961713430826443361839218688, 3092211361843297279446939336290358883803229084188672, 11655448643885766555067489704152609539452036821549056, 43932793458619644930597180632292653357340101122719744, 165595542483919987709589822769971978052339613025959936, 624178012180638076281972068521820991977348887857856512, 2352709408996346450179517267224962091484737803623858176, 8868049587075217864514921854751394497448975621320343552, 33426271505571671142986190867389672021710569018830618624, 125993389616655926913845030260173859704682084721523425280, 474905920166670157240297804015654804995680671868756426752
Generating function in Maple syntax:
(x-1)*(2*x^2-4*x+1)/(6*x^3-10*x^2+6*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(2 x^{2}-4 x +1\right)}{6 x^{3}-10 x^{2}+6 x -1}
Generating function in sympy syntax:
(x - 1)*(2*x**2 - 4*x + 1)/(6*x**3 - 10*x**2 + 6*x - 1)
Implicit equation for the generating function in Maple syntax:
(6*x^3-10*x^2+6*x-1)*F(x)-(x-1)*(2*x^2-4*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(6 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(2 x^{2}-4 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/21888*(133*((I-15/133*19^(1/2))*3^(1/2)-45/133*I*19^(1/2)+1)*2^(2/3)*(67+9*19^(1/2)*3^(1/2))^(2/3)+4864+304*((I+3/19*19^(1/2))*3^(1/2)-9/19*I*19^(1/2)-1)*2^(1/3)*(67+9*19^(1/2)*3^(1/2))^(1/3))*(67/1152*((I-9/67*19^(1/2))*3^(1/2)-27/67*I*19^(1/2)+1)*2^(2/3)*(67+9*19^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(134+18*19^(1/2)*3^(1/2))^(1/3)+1/36*(134+18*19^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/21888*(-304*2^(1/3)*((I-3/19*19^(1/2))*3^(1/2)-9/19*I*19^(1/2)+1)*(67+9*19^(1/2)*3^(1/2))^(1/3)+4864-133*2^(2/3)*((I+15/133*19^(1/2))*3^(1/2)-45/133*I*19^(1/2)-1)*(67+9*19^(1/2)*3^(1/2))^(2/3))*(-67/1152*((I+9/67*19^(1/2))*3^(1/2)-27/67*I*19^(1/2)-1)*2^(2/3)*(67+9*19^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(134+18*19^(1/2)*3^(1/2))^(1/3)+1/36*(134+18*19^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)-1/228*(1/576*2^(2/3)*(9*19^(1/2)*3^(1/2)-67)*(67+9*19^(1/2)*3^(1/2))^(2/3)-1/18*(134+18*19^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)*(2^(1/3)*(19^(1/2)*3^(1/2)-19/3)*(67+9*19^(1/2)*3^(1/2))^(1/3)-152/3+(-5/16*19^(1/2)*3^(1/2)*2^(2/3)+133/48*2^(2/3))*(67+9*19^(1/2)*3^(1/2))^(2/3)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(133 \left(\left(\mathrm{I}-\frac{15 \sqrt{19}}{133}\right) \sqrt{3}-\frac{45 \,\mathrm{I} \sqrt{19}}{133}+1\right) 2^{\frac{2}{3}} \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+4864+304 \left(\left(\mathrm{I}+\frac{3 \sqrt{19}}{19}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{19}}{19}-1\right) 2^{\frac{1}{3}} \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{67 \left(\left(\mathrm{I}-\frac{9 \sqrt{19}}{67}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{19}}{67}+1\right) 2^{\frac{2}{3}} \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{1152}-\frac{\mathrm{I} \sqrt{3}\, \left(134+18 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(134+18 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{21888}\\+\\\frac{\left(-304 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{19}}{19}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{19}}{19}+1\right) \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+4864-133 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{15 \sqrt{19}}{133}\right) \sqrt{3}-\frac{45 \,\mathrm{I} \sqrt{19}}{133}-1\right) \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{67 \left(\left(\mathrm{I}+\frac{9 \sqrt{19}}{67}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{19}}{67}-1\right) 2^{\frac{2}{3}} \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{1152}+\frac{\mathrm{I} \sqrt{3}\, \left(134+18 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(134+18 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{21888}\\-\\\frac{\left(\frac{2^{\frac{2}{3}} \left(9 \sqrt{19}\, \sqrt{3}-67\right) \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{576}-\frac{\left(134+18 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n} \left(2^{\frac{1}{3}} \left(\sqrt{19}\, \sqrt{3}-\frac{19}{3}\right) \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{152}{3}+\left(-\frac{5 \sqrt{19}\, \sqrt{3}\, 2^{\frac{2}{3}}}{16}+\frac{133 \,2^{\frac{2}{3}}}{48}\right) \left(67+9 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right)}{228} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 6*a(n)-10*a(n+1)+6*a(n+2), n >= 4
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(n +3\right) = 6 a \! \left(n \right)-10 a \! \left(n +1\right)+6 a \! \left(n +2\right), \quad n \geq 4
Specification 1
Strategy pack name: insertion_partial_point_placements_expand_verified
Tree: http://permpal.com/tree/24125/
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[5,x]
F[4,x] = x
F[5,x] = F[23,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]+F[16,x]+F[21,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[15,x]+F[16,x]+F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[18,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[10,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[44,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[15,x]+F[19,x]+F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[27,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[15,x]+F[25,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[32,x]
F[35,x] = F[15,x]+F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[31,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = F[15,x]+F[36,x]+F[39,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[32,x]+F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[23,x]+F[46,x]
F[46,x] = F[32,x]+F[47,x]
F[47,x] = F[15,x]+F[36,x]+F[42,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[46,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{44}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{31}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{46}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_23(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_21(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_18(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_10(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_44(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_15(x) + F_19(x) + F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_27(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_15(x) + F_25(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_32(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_31(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_36(x) + F_39(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_32(x) + F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_23(x) + F_46(x))
Eq(F_46(x), F_32(x) + F_47(x))
Eq(F_47(x), F_15(x) + F_36(x) + F_42(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_46(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": true, "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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_partial_point_placements_expand_verified", "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:
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