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 1-3 of 3 results.

A347478 Number of total dominating sets in the n-alkane graph.

Original entry on oeis.org

15, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656, 1125899906842624
Offset: 1

Views

Author

Eric W. Weisstein, Sep 03 2021

Keywords

Crossrefs

Cf. A000302 (powers of 4).

Formula

a(n) = 4^(n+1) for n > 1.

A347501 Number of dominating sets in the n-alkane graph.

Original entry on oeis.org

17, 81, 405, 2025, 10125, 50625, 253125, 1265625, 6328125, 31640625, 158203125, 791015625, 3955078125, 19775390625, 98876953125, 494384765625, 2471923828125, 12359619140625, 61798095703125, 308990478515625, 1544952392578125, 7724761962890625
Offset: 1

Views

Author

Eric W. Weisstein, Sep 04 2021

Keywords

Crossrefs

Programs

  • Mathematica
    Join[{17}, LinearRecurrence[{5}, {81}, 20]]
    CoefficientList[Series[(4 x - 17)/(5 x - 1), {x, 0, 20}], x]

Formula

a(n) = 81*5^(n-2) for n > 1.
G.f.: x*(4*x - 17)/(5*x - 1).
E.g.f.: (81*exp(5*x) + 20*x - 81)/25. - Stefano Spezia, Sep 04 2021

A228603 a(1) = 9, a(2) = 44, a(n) = 4*(a(n-1) + a(n-2)) (n >=3).

Original entry on oeis.org

9, 44, 212, 1024, 4944, 23872, 115264, 556544, 2687232, 12975104, 62649344, 302497792, 1460588544, 7052345344, 34051735552, 164416323584, 793872236544, 3833154240512, 18508105908224, 89365040594944, 431492586012672, 2083430506430464, 10059692369772544
Offset: 1

Views

Author

Emeric Deutsch, Nov 02 2013

Keywords

Comments

a(n) = number of independent vertex subsets (i.e. the Merrifield-Simmons index) of the normal alkyl radical of n carbons (i.e. CH_3(CH_2)_{n-1}).

References

  • R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.

Crossrefs

Cf. A228602.

Programs

  • Maple
    a := proc (n) if n = 1 then 9 elif n = 2 then 44 else 4*a(n-1)+4*a(n-2) end if end proc: seq(a(n), n = 1 .. 25);
  • Mathematica
    LinearRecurrence[{4,4},{9,44},30] (* Harvey P. Dale, Oct 30 2016 *)

Formula

a(n) = (8 - 5*sqrt(2))*(2 - 2*sqrt(2))^(n)/8 + (8 + 5*sqrt(2))*(2 + 2*sqrt(2))^(n)/8.
G.f.: x*(9+8*x)/(1-4*x-4*x^2).
a(n) = 9*A057087(n-1)+8*A057087(n-2). - R. J. Mathar, Nov 24 2013
Showing 1-3 of 3 results.