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 A202736 Number of n X 2 0..1 arrays with row sums equal and column sums unequal to adjacent columns. 6
 2, 2, 8, 10, 32, 44, 128, 186, 512, 772, 2048, 3172, 8192, 12952, 32768, 52666, 131072, 213524, 524288, 863820, 2097152, 3488872, 8388608, 14073060, 33554432, 56708264, 134217728, 228318856, 536870912, 918624304, 2147483648, 3693886906 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Column 2 of A202742. a(n) is the number of binary words of length n such that the number of 0's is not equal to the number of 1's. - Geoffrey Critzer, Dec 05 2013 Also the degree of the irreducible polynomial that defines the multifocal ellipsoid with n foci, see links. - Moritz Firsching, Aug 31 2015 LINKS R. H. Hardin, Table of n, a(n) for n = 1..210 Jiawang Nie, Pablo A. Parrilo, Bernd Sturmfels, Semidefinite Representation of the k-Ellipse, arXiv:math/0702005 [math.AG], 2007. FORMULA For odd n, a(n) = 2^n, for even n, a(n) = 2^n - binomial(n,n/2). - Geoffrey Critzer, Dec 05 2013 a(n) = 2^n*(1-(((-1)^n+1)*Gamma((n+1)/2))/(2*sqrt(Pi)*Gamma((n+2)/2))). - Peter Luschny, Sep 10 2014 a(n) = 2^n - A126869(n). - Peter Luschny, Sep 10 2014 From Robert Israel, Aug 31 2015: (Start) G.f.: 1/(1-2*x) - 1/sqrt(1-4*x^2). E.g.f.: exp(2*x) - I_0(2*x) where I_0 is a modified Bessel function. a(n) = ((-8*n+16)*a(n-3)+(4*n-4)*a(n-2)+(2*n-2)*a(n-1))/n. (End) EXAMPLE Some solutions for n=5 ..0..1....0..1....0..1....0..1....0..1....1..0....0..1....1..0....0..1....0..1 ..0..1....0..1....0..1....1..0....1..0....1..0....0..1....1..0....0..1....1..0 ..1..0....1..0....0..1....1..0....0..1....0..1....1..0....1..0....0..1....0..1 ..0..1....1..0....1..0....1..0....0..1....1..0....0..1....1..0....0..1....1..0 ..0..1....1..0....1..0....1..0....0..1....1..0....1..0....1..0....1..0....1..0 MAPLE seq(2^n - `if`(n::even, binomial(n, n/2), 0), n = 1 .. 30); # Robert Israel, Aug 31 2015 MATHEMATICA f[n_]:= If[EvenQ[n], 2^n-Binomial[n, n/2], 2^n]; Drop[Table[f[n], {n, 0, 32}], 1] (* Geoffrey Critzer, Dec 05 2013 *). PROG (Sage) A202736 = lambda n: 2^n*(1-(((-1)^n+1)*gamma((n+1)/2))/(2*sqrt(pi)*gamma((n+2)/2))) [A202736(n) for n in (1..32)] # Peter Luschny, Sep 10 2014 (Magma) I:=[2, 2, 8]; [n le 3 select I[n] else ((-8*n+16)*Self(n-3)+(4*n-4)*Self(n-2)+(2*n-2)*Self(n-1))/n: n in [1..40]]; // Vincenzo Librandi, Sep 01 2015 CROSSREFS Cf. A126869. Sequence in context: A320138 A325102 A275436 * A179989 A046982 A015620 Adjacent sequences: A202733 A202734 A202735 * A202737 A202738 A202739 KEYWORD nonn AUTHOR R. H. Hardin, Dec 23 2011 STATUS approved

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Last modified August 6 13:30 EDT 2024. Contains 374974 sequences. (Running on oeis4.)