%I #31 Feb 21 2022 13:01:06

%S 1,2,4,8,16,32,64,127,247,487,961,1897,3745,7393,14593,28801,56833,

%T 112156,221341,436825,862094,1701380,3357739,6626611,13077820,

%U 25809478,50935832,100523529,198386490,391522260,772682018,1524913233,3009466064,5939279536,11721362180

%N 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.

%C 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<ell.

%H Georg Fischer, <a href="/A335247/b335247.txt">Table of n, a(n) for n = 0..1000</a>

%H Kees Immink and Kui Cai, <a href="https://www.researchgate.net/publication/341263022">Properties and constructions of energy-harvesting sliding-window constrained code</a>, IEEE Communications Letters, May 2020.

%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, 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).

%F 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).

%F 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.

%F a(n) ~ c*r^n where c = 1.07317641333 and r = 1.9735326811117101072.

%t 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 *)

%t 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 *)

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

%K nonn,easy

%O 0,2

%A _Kees Immink_, May 28 2020