A164392 Number of binary strings of length n with no substrings equal to 0001 or 0010.

1, 2, 4, 8, 14, 25, 44, 78, 137, 241, 423, 743, 1304, 2289, 4017, 7050, 12372, 21712, 38102, 66865, 117340, 205918, 361361, 634145, 1112847, 1952911, 3427120, 6014177, 10554145, 18521234, 32502500, 57037912, 100094558, 175653705, 308250764, 540942382

Offset

0,2

Comments

Nonnegative walks with n steps on the x-axis starting at the origin using steps {1,0,-1} and visiting no point more than twice. Note: a 0 step counts as a visit and a step but does not contribute to the length of the walk. - David Scambler, May 22 2012

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 500 terms from R. H. Hardin)

Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 1, -1).

Formula

From David Scambler, May 22 2012: (Start)

G.f.: (1+x^3-x^4)/(1-2*x+x^3-x^4+x^5).

a(n) = 2^n for n<4; otherwise, a(n) = a(n-1)+a(n-2)+a(n-4)+1. (End)

Mathematica

CoefficientList[Series[ (1+x^3-x^4)/(1-2*x+x^3-x^4+x^5) , {x, 0, 45}], x] (* David Scambler, May 22 2012 *)

Prog

(PARI) x='x+O('x^50); Vec( (1+x^3-x^4)/(1-2*x+x^3-x^4+x^5) ) \\ G. C. Greubel, sep 18 2017

Crossrefs

Cf. A212584, A212585, A212586, A212587, A212589.

Sequence in context: A164391 A164153 A212588 * A164152 A164390 A164151

Adjacent sequences: A164389 A164390 A164391 * A164393 A164394 A164395

Keyword

nonn,easy,walk

Author

R. H. Hardin, Aug 14 2009

Extensions

Edited by Alois P. Heinz, Oct 27 2017

Status

approved