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, 8589934592, 14846262964

Offset: 1

Author

R. H. Hardin, Dec 23 2011

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A126869.

Programs

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

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,0,34}],1] (* Geoffrey Critzer, Dec 05 2013 *)

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

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)