A185828 - OEIS

%I #32 Jan 26 2020 14:50:13

%S 1,3,10,23,61,162,421,1103,2890,7563,19801,51842,135721,355323,930250,

%T 2435423,6376021,16692642,43701901,114413063,299537290,784198803,

%U 2053059121,5374978562,14071876561,36840651123,96450076810,252509579303

%N Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

%C Column 2 of A185835.

%H R. H. Hardin, <a href="/A185828/b185828.txt">Table of n, a(n) for n = 1..200</a>

%F Empirical: a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).

%F a(n) = n*Sum_{k=0..n-1} C(2n-2k, 2k)/(n-k). - _Paul D. Hanna_, Mar 19 2011

%F L.g.f.: Sum_{n>=1} a(n)*x^n/n = -log((1+x+x^2)*(1-3*x+x^2))/2. - _Paul D. Hanna_, Mar 19 2011

%F Logarithmic derivative of A051286, which is the Whitney number of level n of the lattice of the ideals of the fence of order 2n. - _Paul D. Hanna_, Mar 19 2011

%F Empirical g.f.: x*(1+x+3*x^2-2*x^3)/(1+x+x^2)/(1-3*x+x^2). - _Colin Barker_, Feb 22 2012

%F Empirical: a(n) = Sum_{k=0..floor(n/2)} A084534(n, 2*k). - _Johannes W. Meijer_, Jun 17 2018

%F Empirical: a(n) = A100886(2n). - _Wojciech Florek_, Jan 26 2020

%e Some solutions for 4 X 2 with a(1,1)=0:

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

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

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

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

%e The logarithmic g.f. begins:

%e L(x) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 162*x^6/6 + ..., where

%e exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 63*x^6 + ... + A051286(n)*x^n/n + ... - _Paul D. Hanna_, Mar 19 2011

%p a := proc(n): n*add(binomial(2*n-2*k, 2*k)/(n-k), k=0..n-1) end: seq(a(n), n=1..28); # _Johannes W. Meijer_, Jun 18 2018

%o (PARI) {a(n)=n*sum(k=0, n-1, binomial(2*n-2*k, 2*k)/(n-k))} /* _Paul D. Hanna_, Mar 19 2011 */

%o (PARI) {a(n)=n*polcoeff(-log( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n))/2, n)} /* _Paul D. Hanna_, Mar 19 2011 */

%Y Cf. A051286 (exp), A180662 (Fi1).

%K nonn

%O 1,2

%A _R. H. Hardin_, Feb 05 2011