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Number of binary arrays of length 2*n+1 with fewer than n ones in any length 2n subsequence (=less than 50% duty cycle).
1

%I #16 Oct 29 2012 07:07:59

%S 1,7,34,151,646,2710,11236,46231,189214,771442,3136156,12720982,

%T 51507964,208260556,841065544,3393346711,13679459854,55106773786,

%U 221860011244,892741834546,3590659699444,14436037598836,58018598086264

%N Number of binary arrays of length 2*n+1 with fewer than n ones in any length 2n subsequence (=less than 50% duty cycle).

%C Row 2 of A213118.

%H R. H. Hardin, <a href="/A213119/b213119.txt">Table of n, a(n) for n = 1..210</a>

%F Recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 8*(2*n-3)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012

%F G.f.: 1/(1-4*x)-3/(2*sqrt(1-4*x)). - _Vaclav Kotesovec_, Oct 21 2012

%F a(n) = 4^n - 3*C(2*n-1,n). - _Vaclav Kotesovec_, Oct 29 2012

%e Some solutions for n=3

%e ..0....0....1....1....0....1....1....0....1....1....0....0....0....0....0....0

%e ..0....1....0....0....0....0....0....0....0....0....1....0....0....0....0....0

%e ..0....0....0....0....0....0....0....1....0....0....1....1....0....0....1....0

%e ..1....0....0....0....0....0....0....0....1....0....0....1....1....1....0....0

%e ..0....1....1....0....0....0....0....0....0....1....0....0....1....0....0....0

%e ..1....0....0....1....1....0....1....0....0....0....0....0....0....0....0....1

%e ..0....0....0....0....1....1....1....0....0....1....0....0....0....0....1....0

%t Table[4^n-3*Binomial[2*n-1,n],{n,1,20}] (* _Vaclav Kotesovec_, Oct 29 2012 *)

%K nonn

%O 1,2

%A _R. H. Hardin_, Jun 05 2012