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A120118
a(n) is the number of binary strings of length n such that no subsequence of length 5 or less contains 3 or more ones.
6
1, 2, 4, 7, 11, 16, 26, 43, 71, 116, 186, 300, 487, 792, 1287, 2087, 3382, 5484, 8898, 14438, 23423, 37993, 61625, 99965, 162165, 263065, 426736, 692229, 1122903, 1821538, 2954849, 4793266, 7775472, 12613097, 20460538, 33190414, 53840404
OFFSET
0,2
FORMULA
a(n) = a(n-1) + a(n-3) + 2*a(n-5) - a(n-8) - a(n-10).
G.f.: 1 + x*(1+x+x^2)*(2+x^2+x^3-x^4-x^5-x^7)/(1-x-x^3-2*x^5+x^8+x^10). - R. J. Mathar, Nov 28 2011
EXAMPLE
This sequence is similar to A118647 - where no subsequence of length 4 contains 3 ones. It is obvious that the first 4 terms of these two sequences are the same. There are only 3 sequences of length 5 that contain 3 ones such that no subsequence of length 4 contains 3 ones: 10101, 11001, 10011. Hence the fifth term for this sequence is 3 less than the corresponding term of A118647.
MATHEMATICA
LinearRecurrence[{1, 0, 1, 0, 2, 0, 0, -1, 0, -1}, {1, 2, 4, 7, 11, 16, 26, 43, 71, 116, 186}, 50] (* Harvey P. Dale, Nov 27 2013 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1 +x*(1 +x+x^2)*(2+x^2+x^3-x^4-x^5-x^7)/(1-x-x^3-2*x^5+x^8+x^10) )); // G. C. Greubel, May 05 2023
(SageMath)
def A120118_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1 +x*(1+x+x^2)*(2+x^2+x^3-x^4-x^5-x^7)/(1-x-x^3-2*x^5 +
x^8+x^10) ).list()
A120118_list(40) # G. C. Greubel, May 05 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova, Aug 15 2006, Oct 11 2006
STATUS
approved