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A334251 a(n) is the number of binary (0,1) sequences of length n that have at most two zeros in a window of seven consecutive symbols.

%I A334251 #50 May 30 2020 19:50:00
%S A334251 1,2,4,7,11,16,22,29,43,66,102,157,239,358,526,777,1159,1740,2619,
%T A334251 3942,5923,8870,13259,19822,29667,44451,66641,99912,149745,224338,
%U A334251 335993,503199,753720,1129164,1691796,2534807,3797721,5689507,8523275,12768309,19127928,28655867,42930562
%N A334251 a(n) is the number of binary (0,1) sequences of length n that have at most two zeros in a window of seven consecutive symbols.
%C A334251 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. We investigate binary systems emitting 0's and 1's signals where it is assumed that the 1's carry the energy. 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 A334251 Kees Immink, <a href="/A334251/b334251.txt">Table of n, a(n) for n = 0..50</a>
%F A334251 G.f.: (x^20 +x^19 +x^18 +2*x^17 +2*x^16 +x^15 -3*x^13 -4*x^12 -5*x^11 -7*x^10 -5*x^9 -3*x^8 -3*x^7 +2*x^6 +3*x^5 +3*x^4 +3*x^3 +2*x^2 +x +1) / (-x^21 -x^18 +x^15 +3*x^14 +x^12 +2*x^11 -3*x^7 -x^4 -x +1).
%F A334251 From _David A. Corneth_, Apr 21 2020: (Start)
%F A334251 a(n) ~ c*r^n where c = 1.81880731105 and r = 1.498122533939865577.
%F A334251 a(n) = a(n - 1) + a(n - 4) + 3*a(n - 6) - 2*a(n - 10) - a(n - 12) - 3*a(n - 13) - a(n - 15) + a(n - 18) + a(n - 21), n >= 21. (End)
%e A334251 a(3) = 7 as there are 8 possible binary (0,1) sequences of length 3 but exactly one of them has more than 2 zero's in a window of seven consecutive symbols (the sequence (000)) leaving 8-1 = 7 such sequences. - _David A. Corneth_, Apr 20 2020
%Y A334251 Cf. A118647, A120118, A133523.
%K A334251 nonn,easy
%O A334251 0,2
%A A334251 _Kees Immink_, Apr 20 2020