A234591 Number of binary words of length n which have no 0^b 1 1 0^a 1 0 1 0^b - matches, where a=1, b=2.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2044, 4084, 8160, 16304, 32576, 65088, 130048, 259841, 519173, 1037329, 2072626, 4141192, 8274272, 16532336, 33032288, 65999871, 131870458, 263482601, 526449078, 1051866919, 2101673384, 4199229896, 8390234112

Offset: 0

Author

N. J. A. Sloane, Jan 01 2014

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• B. K. Miceli, J, Remmel, Minimal Overlapping Embeddings and Exact Matches in Words, PU. M. A., Vol. 23 (2012), No. 3, pp. 291-315.
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,0,-1,1,1).

Programs

GAP

a:=[1,2,4,8,16,32,64,128,256,512];; for n in [11..40] do a[n]:=2*a[n-1]-a[n-8]+a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 14 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^8+x^9)/(1-2*x+x^8-x^9-x^10) )); // G. C. Greubel, Sep 14 2019

Maple

a:= n-> coeff(series(-(x^9+x^8+1)/(x^10+x^9-x^8+2*x-1), x, n+1), x, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 08 2014

Mathematica

a[n_ /; n<=9]:= 2^n; a[n_]:=a[n] =2*a[n-1] -a[n-8] +a[n-9] +a[n-10]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Mar 18 2014 *)

PARI

my(x='x+O('x^40)); Vec((1+x^8+x^9)/(1-2*x+x^8-x^9-x^10)) \\ G. C. Greubel, Sep 14 2019

Sage

def A234591_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^8+x^9)/(1-2*x+x^8-x^9-x^10)).list()
A234591_list(40) # G. C. Greubel, Sep 14 2019

Formula

G.f.: (1+x^8+x^9)/(1-2*x+x^8-x^9-x^10). - Alois P. Heinz, Jan 08 2014

Extensions

a(17)-a(33) from Alois P. Heinz, Jan 08 2014