cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A286945 Number of maximal matchings in the ladder graph P_2 X P_n.

Original entry on oeis.org

1, 2, 5, 11, 24, 51, 109, 234, 503, 1081, 2322, 4987, 10711, 23006, 49415, 106139, 227976, 489669, 1051759, 2259072, 4852259, 10422163, 22385754, 48082339, 103276009, 221826440, 476460797, 1023389687, 2198137722, 4721377893, 10141043023, 21781936530
Offset: 1

Views

Author

Andrew Howroyd, May 16 2017

Keywords

Links

Crossrefs

Cf. A284703, A284710, A286911.

Programs

  • GAP
    a:=[1,2,5,11,24];; for n in [6..35] do a[n]:=2*a[n-1]+a[n-4]+a[n-5]; od; a; # G. C. Greubel, Dec 30 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5) )); // G. C. Greubel, Dec 30 2019
  • Maple
    seq(coeff(series(x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5), x, n+1), x, n), n = 1..35); # G. C. Greubel, Dec 30 2019
  • Mathematica
    Table[3Cos[nPi/3]/13 - 5Sin[nPi/3]/(13 Sqrt[3]) + RootSum[-1 -2# -#^2 +#^3 &, (-6 -72# +80#^2) #^n &]/403, {n, 35}] (* Eric W. Weisstein, Jul 13 2017 *)
LinearRecurrence[{2,0,0,1,1}, {1,2,5,11,24}, 35] (* Eric W. Weisstein, Jul 13 2017 *)
CoefficientList[Series[(1+x^2+x^3+x^4)/(1-2x-x^4-x^5), {x, 0, 35}], x] (* Eric W. Weisstein, Jul 13 2017 *)
  • PARI
    Vec((1+x^2+x^3+x^4)/((1-x+x^2)*(1-x-2*x^2-x^3)) + O(x^35))
  • Sage
    def A286945_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5) ).list()
a=A286945_list(35); a[1:] # G. C. Greubel, Dec 30 2019

Formula

a(n) = 2*a(n-1) + a(n-4) + a(n-5) for n>5.
G.f.: x*(1+x^2+x^3+x^4)/((1-x+x^2)*(1-x-2*x^2-x^3)).

A284703 Number of maximal matchings in the n-prism graph.

Original entry on oeis.org

1, 5, 10, 17, 51, 98, 211, 457, 964, 2095, 4489, 9638, 20723, 44469, 95550, 205225, 440777, 946808, 2033571, 4367947, 9381928, 20151345, 43283195, 92967814, 199685501, 428904403, 921243124, 1978737477, 4250128177, 9128846128, 19607840133, 42115660425
Offset: 1

Views

Author

Eric W. Weisstein, Apr 01 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 16 2017

Links

Crossrefs

Cf. A284701, A286945, A284710, A102080, A068397, A123304.

Programs

  • Magma
    I:=[1,5,10,17,51,98,211,457]; [n le 8 select I[n] else Self(n-1)+2*Self(n-2)+Self(n-3)-Self(n-4)+2*Self(n-5)+Self(n-6)-Self(n-7)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, May 17 2017
  • Mathematica
    LinearRecurrence[{1, 2, 1, -1, 2, 1, -1, -1}, {1, 5, 10, 17, 51, 98, 211, 457}, 40] (* Vincenzo Librandi, May 17 2017 *)
CoefficientList[Series[(-8 x^7 - 7 x^6 + 6 x^5 + 10 x^4 - 4 x^3 + 3 x^2 + 4 x + 1) / ((x^2 - x + 1) (x^3 - x - 1) (x^3 + 2 x^2 + x - 1)), {x, 0, 33}], x] (* Vincenzo Librandi, May 17 2017 *)
Table[2 Cos[n Pi/3] + RootSum[-1 - 2 # - #^2 + #^3 &, #^n &] +
  RootSum[-1 + #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 17 2017 *)
  • PARI
    Vec((-8*x^7-7*x^6+6*x^5+10*x^4-4*x^3+3*x^2+4*x+1)/((x^2-x+1)*(x^3-x-1)*(x^3+2*x^2+x-1))+O(x^20)) \\ Andrew Howroyd, May 16 2017

Formula

From Andrew Howroyd, May 16 2017 (Start)
a(n) = a(n-1)+2*a(n-2)+a(n-3)-a(n-4)+2*a(n-5)+a(n-6)-a(n-7)-a(n-8) for n>8.
G.f.: x*(-8*x^7-7*x^6+6*x^5+10*x^4 -4*x^3+3*x^2+4*x+1)/((x^2-x+1)*(x^3-x-1)*(x^3+2*x^2+x-1)).
(End)

Extensions

a(1)-a(2) and a(20)-a(32) from Andrew Howroyd, May 16 2017

A284701 Number of maximal matchings in the n-antiprism graph.

Original entry on oeis.org

2, 6, 14, 46, 137, 354, 905, 2366, 6278, 16681, 44156, 116650, 308180, 814645, 2153984, 5695102, 15056494, 39804582, 105231559, 278204561, 735502187, 1944477640, 5140687360, 13590620330, 35930023287, 94989547620, 251127430313, 663914974741
Offset: 1

Views

Author

Eric W. Weisstein, Apr 01 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 16 2017

Links

Crossrefs

Cf. A284703, A284710, A192742, A284700.

Programs

  • Mathematica
    LinearRecurrence[{2, 1, 0, 3, 5, 1, -2, -1}, {2, 6, 14, 46, 137, 354,
  905, 2366}, 20] (* Eric W. Weisstein, May 17 2017 *)
CoefficientList[Series[x*(-8*x^7-14*x^6+6*x^5+25*x^4+12*x^3+2*x+2)/(x^8 +2*x^7-x^6-5*x^5 -3*x^4-x^2-2*x+1), {x, 0, 50}], x] (* G. C. Greubel, May 17 2017 *)
Table[RootSum[1 + 2 # - #^2 - 5 #^3 - 3 #^4 - #^6 - 2 #^7 + #^8 &, #^n &], {n, 10}] (* Eric W. Weisstein, May 26 2017 *)
  • PARI
    Vec((-8*x^7-14*x^6+6*x^5+25*x^4+12*x^3+2*x+2)/(x^8+2*x^7-x^6-5*x^5-3*x^4-x^2-2*x+1)+O(x^20)) \\ Andrew Howroyd, May 16 2017

Formula

From Andrew Howroyd, May 16 2017 (Start)
a(n) = 2*a(n-1) + a(n-2) + 3*a(n-4) + 5*a(n-5) + a(n-6) - 2*a(n-7) - a(n-8) for n>8.
G.f.: x*(-8*x^7 - 14*x^6 + 6*x^5 + 25*x^4 + 12*x^3 + 2*x + 2)/(x^8 + 2*x^7 - x^6 - 5*x^5 - 3*x^4 - x^2 - 2*x + 1). (End)

Extensions

a(1)-a(2) and a(16)-a(28) from Andrew Howroyd, May 16 2017

A290470 Number of minimal edge covers in the n-Moebius ladder.

Original entry on oeis.org

3, 7, 15, 59, 143, 367, 1039, 2755, 7395, 20007, 53727, 144635, 389535, 1048159, 2821535, 7595267, 20443523, 55029319, 148125295, 398712379, 1073232175, 2888862159, 7776059055, 20931132355, 56341155043, 151655701607, 408217663167, 1098815603707, 2957725352255
Offset: 1

Views

Author

Eric W. Weisstein, Aug 03 2017

Keywords

Links

Crossrefs

Cf. A020866, A020877, A020878, A284710.

Programs

  • Mathematica
    Table[2 Cos[n Pi/2] - RootSum[-1 + # + #^2 + #^3 &, #^n &] +
  RootSum[1 - 2 #^2 - 2 #^3 + #^4 &, #^n &], {n, 20}]
LinearRecurrence[{1, 2, 6, 2, 2, -2, -2, -1, 1}, {3, 7, 15, 59, 143, 367, 1039, 2755, 7395}, 20]
CoefficientList[Series[-(((1 + x) (-3 - x - x^2 + x^3) (-1 - 4 x^3 + 3 x^4))/((1 + x^2) (-1 - x - x^2 + x^3) (1 - 2 x - 2 x^2 + x^4))), {x, 0, 20}], x]
  • PARI
    Vec((1 + x)*(1 + 4*x^3 - 3*x^4)*(3 + x + x^2 - x^3)/((1 + x^2)*(1 + x + x^2 - x^3)*(1 - 2*x - 2*x^2 + x^4)) + O(x^30)) \\ Andrew Howroyd, Aug 04 2017

Formula

From Andrew Howroyd, Aug 04 2017: (Start)
a(n) = a(n-1) + 2*a(n-2) + 6*a(n-3) + 2*a(n-4) + 2*a(n-5) - 2*a(n-6) - 2*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(1 + x)*(1 + 4*x^3 - 3*x^4)*(3 + x + x^2 - x^3)/((1 + x^2)*(1 + x + x^2 - x^3)*(1 - 2*x - 2*x^2 + x^4)).
(End)

Extensions

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 04 2017
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.