A234592 Number of binary words of length n which have no 0^b 1 1 0^a 1 0 1 0^b - matches, where a=b=2.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2047, 4092, 8180, 16352, 32688, 65344, 130624, 261120, 521984, 1043457, 2085893, 4169745, 8335410, 16662664, 33309024, 66585456, 133105760, 266081280, 531902207, 1063283962, 2125527529, 4248975286, 8493793063

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 (2,0,0,0,0,0,0,0,-1,1,1).

Programs

GAP

a:=[1,2,4,8,16,32,64,128,256,512,1024];; for n in [12..40] do a[n]:=2*a[n-1]-a[n-9]+a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 14 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^9+x^10)/(1-2*x+x^9-x^10-x^11) )); // G. C. Greubel, Sep 14 2019

Maple

a:= n-> coeff(series(-(x^10+x^9+1)/(x^11+x^10-x^9+2*x-1), x, n+1), x, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 08 2014

Mathematica

a[n_ /; n<=10]:= 2^n; a[n_]:=a[n] =2*a[n-1] -a[n-9] +a[n-10] +a[n-11]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 18 2014 *)
LinearRecurrence[{2,0,0,0,0,0,0,0,-1,1,1}, {1,2,4,8,16,32,64,128,256, 512,1024}, 40] (* Harvey P. Dale, May 17 2018 *)

PARI

my(x='x+O('x^40)); Vec((1+x^9+x^10)/(1-2*x+x^9-x^10-x^11)) \\ G. C. Greubel, Sep 14 2019

Sage

def A234592_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^9+x^10)/(1-2*x+x^9-x^10-x^11)).list()
A234592_list(40) # G. C. Greubel, Sep 14 2019

Formula

G.f.: (1+x^9+x^10)/(1-2*x+x^9-x^10-x^11). - Alois P. Heinz, Jan 08 2014

Extensions

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