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

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

Offset: 0

Author

David Kofoed Wind, Aug 16 2012

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 0..700
• Index entries for linear recurrences with constant coefficients, signature (52,-677,26).

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).