A334251 -id:A334251 - cp's OEIS frontend

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.

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

Tanya Khovanova, Aug 15 2006, Oct 11 2006

			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.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kees Immink and Kui Cai, Properties and constructions of energy-harvesting sliding-window constrained codes, IEEE Comm. Lett., May 2020.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,2,0,0,-1,0,-1).

Cf. A118647, A133523, A133551, A334251, A335247.

R:=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

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 *)

def A120118_list(prec):
    P. = 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

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

A335247 a(n) is the number of binary (0,1) sequences of length n that have at least two ones in each window of eight consecutive symbols.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 127, 247, 487, 961, 1897, 3745, 7393, 14593, 28801, 56833, 112156, 221341, 436825, 862094, 1701380, 3357739, 6626611, 13077820, 25809478, 50935832, 100523529, 198386490, 391522260, 772682018, 1524913233, 3009466064, 5939279536, 11721362180

Offset: 0

Kees Immink, May 28 2020

Application: Not all electronic devices connected to the Internet of Things (IoT) have batteries or are connected to the power cable. These self-contained devices must rely on the harvesting of energy of the signals sent by a transmitter. A minimal number of 1's in transmitted sequences is required so as to carry sufficient energy within a prescribed time span. A binary sequence is said to obey the sliding-window (ell,t)-constraint if the number of 1's within any window of ell consecutive bits of that sequence is at least t, t

Georg Fischer, Table of n, a(n) for n = 0..1000
Kees Immink and Kui Cai, Properties and constructions of energy-harvesting sliding-window constrained code, IEEE Communications Letters, May 2020.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,0,1,2,3,0,-3,-4,-3,0,0,-2,-3,0,0,3,3,0,0,0,1,0,0,0,-1).

Cf. A118647, A120118, A133523, A334251, A133551.

CoefficientList[Series[-(x^27 +x^26 -x^23 -x^22 -3*x^19 -5*x^18 -3*x^17 +3*x^15 +4*x^14 +2*x^13 +3*x^11 +5*x^10 +5*x^9 +3*x^8 -3*x^7 -3*x^6 -2*x^5 -x^4 -x^3 -x^2 -x -1) / (x^28 -x^24 -3*x^20 -3*x^19 +3*x^16 +2*x^15 +3*x^12 +4*x^11 +3*x^10 -3*x^8 -2*x^7 -x^6 -x^4 -x^3 -x^2 -x +1),{x,0,100}],x] (* Georg Fischer, Oct 26 2020 *)
LinearRecurrence[{1,1,1,1,0,1,2,3,0,-3,-4,-3,0,0,-2,-3,0,0,3,3,0,0,0,1,0,0,0,-1},{1,2,4,8,16,32,64,127,247,487,961,1897,3745,7393,14593,28801,56833,112156,221341,436825,862094,1701380,3357739,6626611,13077820,25809478,50935832,100523529},40] (* Harvey P. Dale, Feb 21 2022 *)

G.f.: -(x^27+x^26-x^23-x^22-3*x^19-5*x^18-3*x^17+3*x^15+4*x^14+2*x^13 +3*x^11 +5*x^10+5*x^9+3*x^8-3*x^7-3*x^6-2*x^5-x^4-x^3-x^2-x-1) / (x^28-x^24-3*x^20 -3*x^19 +3*x^16 +2*x^15+3*x^12+4*x^11+3*x^10-3*x^8-2*x^7-x^6-x^4-x^3-x^2-x+1).
a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-6)+2*a(n-7)+3*a(n-8)-3*a(n-10) -4*a(n-11) -3*a(n-12) -2*a(n-15)-3*a(n-16)+3*a(n-19)+3*a(n-20)+a(n-24)-a(n-28), n>28.
a(n) ~ c*r^n where c = 1.07317641333 and r = 1.9735326811117101072.

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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.

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