cp's OEIS Frontend

A185828 Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

%I A185828 #32 Jan 26 2020 14:50:13
%S A185828 1,3,10,23,61,162,421,1103,2890,7563,19801,51842,135721,355323,930250,
%T A185828 2435423,6376021,16692642,43701901,114413063,299537290,784198803,
%U A185828 2053059121,5374978562,14071876561,36840651123,96450076810,252509579303
%N A185828 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 A185828 Column 2 of A185835.
%H A185828 R. H. Hardin, <a href="/A185828/b185828.txt">Table of n, a(n) for n = 1..200</a>
%F A185828 Empirical: a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
%F A185828 a(n) = n*Sum_{k=0..n-1} C(2n-2k, 2k)/(n-k). - _Paul D. Hanna_, Mar 19 2011
%F A185828 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 A185828 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 A185828 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 A185828 Empirical: a(n) = Sum_{k=0..floor(n/2)} A084534(n, 2*k). - _Johannes W. Meijer_, Jun 17 2018
%F A185828 Empirical: a(n) = A100886(2n). - _Wojciech Florek_, Jan 26 2020
%e A185828 Some solutions for 4 X 2 with a(1,1)=0:
%e A185828   0 0   0 1   0 0   0 0   0 1   0 0   0 0   0 0   0 0   0 0
%e A185828   1 1   0 1   0 1   1 1   0 1   1 0   0 1   1 1   1 0   0 1
%e A185828   0 1   0 0   0 1   0 1   1 0   1 0   1 1   1 1   1 1   0 1
%e A185828   0 0   1 1   0 0   0 1   1 0   0 0   0 0   0 0   0 0   0 1
%e A185828 The logarithmic g.f. begins:
%e A185828 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 A185828 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 A185828 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 A185828 (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 A185828 (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 A185828 Cf. A051286 (exp), A180662 (Fi1).
%K A185828 nonn
%O A185828 1,2
%A A185828 _R. H. Hardin_, Feb 05 2011