A202736 Number of n X 2 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

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

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