This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181296 #28 May 12 2025 10:14:03
%S A181296 0,2,10,48,208,864,3472,13640,52664,200616,755992,2823688,10468856,
%T A181296 38570504,141341944,515532424,1872673144,6777925768,24453094264,
%U A181296 87966879368,315629269368,1129834372744,4035747287416,14387491636872
%N A181296 The number of odd entries in all the 2-compositions of n.
%C A181296 Also number of columns with distinct entries in all compositions of n.
%C A181296 A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
%H A181296 G. Castiglione, A. Frosini, E. Munarini, A. Restivo, and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European Journal of Combinatorics, 28, 2007, 1724-1741.
%H A181296 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-12,-4,12,-4).
%F A181296 G.f.: 2*z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2).
%F A181296 a(n) = Sum_{k=0..n} k*A181295(n,k) = Sum_{k=0..n} k*A181302(n,k).
%F A181296 a(n) = 2*A181305(n). - _R. J. Mathar_, Oct 28 2010
%F A181296 a(n) = 7*a(n-1)- 12*a(n-2)- 4*a(n-3)+12*a(n-4)-4*a(n-5). - _Harvey P. Dale_, Nov 11 2011
%F A181296 E.g.f.: exp(-x)*(exp(3*x)*(16*(1 + 7*x)*cosh(sqrt(2)*x) + sqrt(2)*(18 + 77*x)*sinh(sqrt(2)*x)) - 16)/98. - _Stefano Spezia_, May 11 2025
%e A181296 a(2) = 10 because in the 2-compositions of 2, namely (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), and (0,0/1,1), we have 2+0+0+2+2+2+2=10 odd entries (the 2-compositions are written as (top row / bottom row)).
%e A181296 a(1)=2 because in (0/1) and (1/0) we have a total of 2 columns with distinct entries (the 2-compositions are written as (top row / bottom row)).
%p A181296 g := 2*z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 25);
%t A181296 CoefficientList[Series[(2x (1-x)^2)/((1+x)(1-4x+2x^2)^2),{x,0,30}],x] (* or *) LinearRecurrence[{7,-12,-4,12,-4},{0,2,10,48,208},30] (* _Harvey P. Dale_, Nov 11 2011 *)
%Y A181296 Cf. A181295, A181297, A181298, A181302.
%K A181296 nonn,easy
%O A181296 0,2
%A A181296 _Emeric Deutsch_, Oct 12 2010
%E A181296 Merged with a definition concerning row sums of A181302 - _R. J. Mathar_, Oct 28 2010