A182027 a(n) = number of n-lettered words in the alphabet {1, 2} with as many occurrences of the substring (consecutive subword) [1, 1] as of [2, 2].

1, 2, 2, 2, 4, 6, 12, 20, 40, 70, 140, 252, 504, 924, 1848, 3432, 6864, 12870, 25740, 48620, 97240, 184756, 369512, 705432, 1410864, 2704156, 5408312, 10400600, 20801200, 40116600, 80233200, 155117520, 310235040, 601080390, 1202160780, 2333606220, 4667212440, 9075135300, 18150270600, 35345263800

Offset: 0

Author

N. J. A. Sloane, Apr 07 2012

Keywords

• nonn

Links

• Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207, 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From _N. J. A. Sloane_, Apr 07 2012]

Crossrefs

Apart from initial terms, same as A063886.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [1,2$3][n+1],
      (2*a(n-1)+4*(n-3)*a(n-2))/(n-1))
    end:
seq(a(n), n=0..39);  # Alois P. Heinz, May 11 2024

Formula

G.f.: 1 + x + x*sqrt((1+2*x)/(1-2*x))= 1 + x + x/G(0), where G(k)= 1 - 2*x/(1 + 2*x/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013

E.g.f.: 1 + x - (x*BesselI(1, 2*x)*(2 + Pi*(1 + 2*x)*StruveL(0, 2*x)) - x*(1 + 2*x)*BesselI(0, 2*x)*(2 + Pi*StruveL(1, 2*x)))/2. - Stefano Spezia, May 11 2024