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A118645
Number of binary strings of length n such that there exist three consecutive digits where at least two of them are 1's.
1
0, 0, 1, 4, 10, 23, 51, 109, 228, 471, 964, 1960, 3967, 8003, 16107, 32362, 64941, 130200, 260866, 522415, 1045831, 2093129, 4188408, 8379967, 16764552, 33535872, 67081663, 134177863, 268377031, 536785286, 1073616333, 2147299732
OFFSET
0,4
COMMENTS
We set a(2) = 1 by convention; there is one string of length 2 which has two consecutive 1's, namely 11. This also makes various formulas simpler.
For n>=3, a(n) = 2^n - the sum of all terms in the (n-3)rd power of the 4 X 4 matrix [[1 1 0 0] [0 0 1 0] [0 0 0 1] [1 1 0 0]] because this matrix represents the transitions from the state where the last three bits are 000, 001, 010, 100 to the state after the next bit, always avoiding two 1's out of the last three bits. - Joshua Zucker, Aug 04 2006
Complementary to A048625 which starts 4,6,9,13,19,28,41,60,88,129,189. For n >= 3, a(n) + A048625(n-3) = 2^n. A048625 is a subsequence of A000930, A068921 and A078012. All of them satisfy the recurrence a(n) = a(n-1) + a(n-3). - Tanya Khovanova, Aug 22 2006
FORMULA
a(n) = 3*2^(n-3) + a(n-1) + a(n-3) for n >= 3. - Tanya Khovanova, Aug 22 2006
From R. J. Mathar, Oct 03 2011: (Start)
G.f.: (x^3 + x^2)/(2*x^4 - x^3 + 2*x^2 - 3*x + 1).
G.f.: x^2 * (x+1)/((2*x-1)*(x^3+x-1)).
a(n) = 2^n - A000930(n+2). (End)
EXAMPLE
a(4) = 10 because we have: 0011, 0101, 0110, 0111, 1010, 1011, 1100, 1101, 1110, 1111. - Geoffrey Critzer, Jan 19 2014
MATHEMATICA
nn=31; r=Solve[{s==1+x s+x b, a==x s, b==x a, c==x a+x b+2x c}, {s, a, b, c}]; CoefficientList[Series[c/.r, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 19 2014 *)
LinearRecurrence[{3, -2, 1, -2}, {0, 0, 1, 4}, 40] (* Harvey P. Dale, Dec 15 2014 *)
PROG
(PARI) x='x+O('x^50); concat([0, 0], Vec((x^3 + x^2)/(2*x^4 - x^3 + 2*x^2 - 3*x + 1))) \\ G. C. Greubel, May 02 2017
CROSSREFS
Sequence in context: A266376 A057750 A295059 * A200759 A159347 A137531
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova, May 10 2006
EXTENSIONS
More terms from Joshua Zucker, Aug 04 2006
Edited by Franklin T. Adams-Watters, Sep 30 2011
STATUS
approved