A118647 a(n) is the number of binary strings of length n such that no subsequence of length 4 contains 3 or more ones.

2, 4, 7, 11, 19, 33, 57, 97, 166, 285, 489, 838, 1436, 2462, 4221, 7236, 12404, 21264, 36453, 62491, 107127, 183646, 314822, 539695, 925191, 1586041, 2718927, 4661017, 7990313, 13697676, 23481725, 40254377, 69007488, 118298524, 202797424

Offset

1,1

Comments

Also, 3 ones in a row are not allowed - this additional condition is only relevant for a(3) which has no subsequences of length 4.

For n>=4, a(n) = the sum of all terms in the n-4th power of the 11 X 11 matrix:

[1 1 0 0 0 0 0 0 0 0 0]

[0 0 1 1 0 0 0 0 0 0 0]

[0 0 0 0 1 1 0 0 0 0 0]

[0 0 0 0 0 0 1 0 0 0 0]

[0 0 0 0 0 0 0 1 1 0 0]

[0 0 0 0 0 0 0 0 0 1 0]

[0 0 0 0 0 0 0 0 0 0 1]

[1 1 0 0 0 0 0 0 0 0 0]

[0 0 1 1 0 0 0 0 0 0 0]

[0 0 0 0 1 1 0 0 0 0 0]

[0 0 0 0 0 0 0 1 1 0 0]

because this matrix represents the transitions from the state where the last four bits are 0000, 0001, 0010, 0011, 0100, 0101, 0110, 1000, 1001, 1010, 1100 to the state after the next bit, always avoiding three 1's out of the last four bits. - Joshua Zucker, Aug 04 2006

Motivated by radar research. In the standard model to get a track on a target you have to get at least M detections out of N observations. See page 96 of Minkler and Minkler. I represented detections as ones and non-detections as zeros. Hence this sequence represents non-tracked situations with n observations.

References

G. Minkler and J. Minkler, CFAR: The Principles of Automatic Radar Detection in Clutter, Magellan, Baltimore, 1990.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

Formula

a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6). - suggested by Jon E. Schoenfield

G.f.: x*(2+2*x+x^2-x^4-x^5) / (1-x-x^2-x^4+x^6). - Colin Barker, Oct 01 2014

Mathematica

LinearRecurrence[{1, 1, 0, 1, 0, -1}, {2, 4, 7, 11, 19, 33}, 40] (* Harvey P. Dale, Oct 03 2016 *)

Prog

(PARI) Vec(x*(2+2*x+x^2-x^4-x^5)/(1-x-x^2-x^4+x^6) + O(x^100)) \\ Colin Barker, Oct 01 2014
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(2+2*x+x^2-x^4-x^5)/(1-x-x^2-x^4+x^6) )); // G. C. Greubel, May 05 2023
(SageMath)
def A118647_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(2+2*x+x^2-x^4-x^5)/(1-x-x^2-x^4+x^6) ).list()
a=A118647_list(41); a[1:] # G. C. Greubel, May 05 2023

Crossrefs

Complementary to A118646: a(n) = 2^n - A118646(n).

Sequence in context: A007864 A277271 A192670 * A000802 A236392 A200377

Adjacent sequences: A118644 A118645 A118646 * A118648 A118649 A118650

Keyword

nonn,easy

Author

Tanya Khovanova, May 10 2006, Aug 17 2006

Extensions

More terms from Joshua Zucker, Aug 04 2006

Status

approved