A234589 Expansion of g.f.: (1+x^6+x^7)/(1-2*x+x^6-x^7-x^8).

1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15937, 31749, 63249, 126002, 251016, 500064, 996207, 1984602, 3953641, 7876278, 15690791, 31258536, 62271945, 124055559, 247138286, 492338537, 980816202, 1953940937, 3892559256, 7754593434, 15448376086, 30775607480, 61309875581, 122138964964

Offset: 0

Author

N. J. A. Sloane, Jan 01 2014

Keywords

• nonn
• easy

Comments

a(n) is the number of binary words of length n which have no 00010100-matches.

Links

• G. C. Greubel, 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,-1,1,1).

Crossrefs

Similar to but different from A172317.

Cf. A234590, A234591, A234592.

Programs

GAP

a:=[1,2,4,8,16,32,64,128];; for n in [9..40] do a[n]:=2*a[n-1]-a[n-6]+a[n-7]+a[n-8]; od; a; # G. C. Greubel, Sep 13 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^6+x^7)/(1-2*x+x^6-x^7-x^8) )); // G. C. Greubel, Sep 13 2019

Maple

seq(coeff(series((1+x^6+x^7)/(1-2*x+x^6-x^7-x^8), x, n+1), x, n), n = 0..40); # G. C. Greubel, Sep 13 2019

Mathematica

CoefficientList[Series[(1+x^6+x^7)/(1-2*x+x^6-x^7-x^8), {x,0,40}], x] (* G. C. Greubel, Sep 13 2019 *)
LinearRecurrence[{2,0,0,0,0,-1,1,1},{1,2,4,8,16,32,64,128},40] (* Harvey P. Dale, Aug 31 2023 *)

PARI

my(x='x+O('x^40)); Vec((1+x^6+x^7)/(1-2*x+x^6-x^7-x^8)) \\ G. C. Greubel, Sep 13 2019

Sage

def A234589_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^6+x^7)/(1-2*x+x^6-x^7-x^8)).list()
A234589_list(40) # G. C. Greubel, Sep 13 2019