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.

User: David Kofoed Wind

David Kofoed Wind's wiki page.

David Kofoed Wind has authored 2 sequences.

A359704 Minimum number of spanning trees in a 3-connected graph on n nodes.

Original entry on oeis.org

16, 45, 75, 209, 336, 928, 1445, 3965, 6000, 16555
Offset: 4

Views

Author

David Kofoed Wind, Jan 11 2023

Keywords

Comments

The initial terms were found through generating all minimally 3-connected graphs with n vertices for n up to 13 (the 3-connected graphs with fewest spanning trees are minimally 3-connected).

Examples

			a(4) = 16 which is the number of spanning trees in the complete graph on 4 nodes.

Links

Crossrefs

Cf. A006290 (3-connected graphs), A199676 (minimally 3-connected graphs).

A215595 Number of strings of length n, formed from the 26-letter English alphabet, which contain the substring xy.

Original entry on oeis.org

0, 0, 1, 52, 2027, 70226, 2280825, 71112600, 2155562551, 64005323902, 1870809923477, 54006556365476, 1543466751232275, 43746473462661450, 1231293799939647601, 34451045198171912752, 959005856055827234927, 26576960554539062120726, 733650711461388661963725
Offset: 0

Views

Author

David Kofoed Wind, Aug 16 2012

Keywords

Examples

			For n = 2, the only such string is xy. For n = 3, there are 26 strings of the form *xy and 26 of the form xy*. For n = 4, there are 26^2 of each of the forms xy**, *xy* and **xy, but we double count xyxy, so the answer for n=4 is 3*26^2 - 1 = 2027.

Links

Crossrefs

Cf. A186314 (same problem for ternary strings).

Programs

  • Mathematica
    Join[{0}, CoefficientList[Series[x/(1 - 52*x + 677*x^2 - 26*x^3), {x, 0, 50}], x]] (* G. C. Greubel, Feb 26 2017 *)
  • PARI
    x='x+O('x^50); concat([0,0], Vec(x/(1 - 52*x + 677*x^2 - 26*x^3))) \\ G. C. Greubel, Feb 26 2017

Formula

a(n) = 26*a(n-1) + 26^(n-2) - a(n-2).
a(n) = 52*a(n-1) - 677*a(n-2) + 26*a(n-3). - Charles R Greathouse IV, Aug 16 2012
G.f.: x/(1 - 52*x + 677*x^2 - 26*x^3). - Alexander R. Povolotsky, Aug 16 2012
a(n) = (1/168)*(13 +2*sqrt(42))^(-n)*(-(84+13*sqrt(42))*(13+2*sqrt(42))^(2*n) + 168*(338+52*sqrt(42))^n-84+13*sqrt(42)). - Alexander R. Povolotsky, Aug 16 2012
a(n) = Sum_{j=1..n} (-1)^(j+1) * B(n,j), where B(n,j) is the number of ways to place k occurrences of xy in a string of length n, and then choosing arbitrary letters for the n - 2k remaining positions. B(n,j) = product((n-i),i=j..(2*j-1)) / j! * 26^(n-2*j).

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

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