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A335247
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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.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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