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A235996 Number of length n binary words that contain at least one pair of consecutive 0's followed by (at some point in the word) at least one pair of consecutive 1's. 0
0, 0, 0, 0, 1, 4, 13, 36, 92, 222, 515, 1160, 2555, 5530, 11804, 24916, 52117, 108204, 223273, 458368, 937020, 1908730, 3876615, 7853840, 15878391, 32045814, 64580028, 129983856, 261354937, 525042292, 1054000645, 2114567580, 4240131740, 8498658390, 17028054539 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,3,2).

FORMULA

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

a(n) -2*a(n-1) = A001629(n-2). - R. J. Mathar, May 06 2016

EXAMPLE

a(5) = 4 because we have: 00011, 00110, 00111, 10011.

We also note that words such as 001011 are included in this enumeration because the pair of consecutive 1's need not immediately follow the pair of consecutive 0's.

MATHEMATICA

nn=30; r=Solve[{s==1+x a+x s, a==x s, b==x a+x b+x c, c==x b, d==x c + 2x d}, {s, a, b, c, d}]; CoefficientList[Series[d/.r, {x, 0, nn}], x]

CoefficientList[ Series[ x^4/((1 - 2x)(1 - x - x^2)^2), {x, 0, 34}], x] (* Robert G. Wilson v, Feb 01 2015 *)

LinearRecurrence[{4, -3, -4, 3, 2}, {0, 0, 0, 0, 1}, 40] (* Harvey P. Dale, Jun 23 2017 *)

CROSSREFS

Sequence in context: A079922 A053563 A221882 * A036636 A036643 A000299

Adjacent sequences:  A235993 A235994 A235995 * A235997 A235998 A235999

KEYWORD

nonn,easy

AUTHOR

Geoffrey Critzer, Jan 18 2014

STATUS

approved

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Last modified May 16 04:38 EDT 2022. Contains 353688 sequences. (Running on oeis4.)