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 A131816 #27 Dec 23 2024 14:53:42
%S A131816 1,2,2,4,3,4,8,5,5,8,16,9,7,9,16,32,17,11,11,17,32,64,33,19,15,19,33,
%T A131816 64,128,65,35,23,23,35,65,128,256,129,67,39,31,39,67,129,256,512,257,
%U A131816 131,71,47,47,71,131,257,512,1024,513,259,135,79,63,79,135,259,513,1024
%N A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).
%C A131816 Row sums = A000295: (1, 4, 11, 26, 57, 120, ...).
%C A131816 If we regard the sequence as an infinite square array read by diagonals then it has the formula U(n,k) = (2^n + 2^k)/2 - 1. This appears to coincide with the number of n X k 0..1 arrays colored with only straight tiles, and new values 0..1 introduced in row major order, i.e., no equal adjacent values form a corner. (Fill the array with 0's and 1's. There must never be 3 adjacent identical values making a corner, only same values in a straight line.) Some solutions with n = k = 4 are:
%C A131816 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1
%C A131816 1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1 0 1 0
%C A131816 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1
%C A131816 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1
%C A131816 (Observation from _R. H. Hardin_, cf. link.) - _M. F. Hasler_ and _N. J. A. Sloane_, Feb 26 2013
%H A131816 Reinhard Zumkeller, <a href="/A131816/b131816.txt">Rows n = 0..120 of triangle, flattened</a>
%H A131816 R. H. Hardin, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-February/010856.html">Post to the SeqFan list</a>, Feb 26 2013
%F A131816 T(n,m) = ((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2. - _Roger L. Bagula_, Oct 16 2008
%e A131816 First few rows of the triangle:
%e A131816 1;
%e A131816 2, 2;
%e A131816 4, 3, 4;
%e A131816 8, 5, 5, 8;
%e A131816 16, 9, 7, 9, 16;
%e A131816 32, 17, 11, 11, 17, 32;
%e A131816 64, 33, 19, 15, 19, 33, 64;
%e A131816 128, 65, 35, 23, 23, 35, 65, 128;
%e A131816 ...
%t A131816 Table[Table[((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2, {m, 0, n}], {n, 0, 10}]; Flatten[%] (* _Roger L. Bagula_, Oct 16 2008 *)
%o A131816 (Haskell)
%o A131816 a131816 n k = a131816_tabl !! n !! k
%o A131816 a131816_row n = a131816_tabl !! n
%o A131816 a131816_tabl = map (map (subtract 1)) $
%o A131816 zipWith (zipWith (+)) a130321_tabl a059268_tabl
%o A131816 -- _Reinhard Zumkeller_, Feb 27 2013
%Y A131816 Cf. A130321, A059268, A000012.
%Y A131816 Row sums give A000295(n+2).
%K A131816 nonn,tabl
%O A131816 0,2
%A A131816 _Gary W. Adamson_, Jul 18 2007
%E A131816 Edited by _N. J. A. Sloane_, Jul 01 2008 at the suggestion of _R. J. Mathar_