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 A053218 #37 Aug 27 2025 11:39:59
%S A053218 1,2,3,3,5,8,4,7,12,20,5,9,16,28,48,6,11,20,36,64,112,7,13,24,44,80,
%T A053218 144,256,8,15,28,52,96,176,320,576,9,17,32,60,112,208,384,704,1280,10,
%U A053218 19,36,68,128,240,448,832,1536,2816,11,21,40,76,144,272,512,960,1792,3328
%N A053218 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1,k-1) for k >= 2 with T(n,1) = n.
%C A053218 Last term in each row gives A001792. Difference between center term of row 2n-1 and row sum of row n, (A053220(n+4) - A053221(n+4)) gives A045618(n).
%C A053218 For all integers k >= 2, if a sequence k,k-1,k+2,k-3,k+4,...,2,2k-2,1,2k-1, b0(n) with offset 1, is written, the sequence b0(2)-b0(1), b0(3)-b0(2), b0(4)-b0(3), ..., b0(2k-1)-b0(2k-2), b1(n) with offset 1, is written under it, the sequence b1(2)-b1(1), b1(3)-b1(2), b1(4)-b1(3), ..., b1(2k-2)-b1(2k-3), b2(n) with offset 1, is written under this, and so on until the sequence b(2k-3)(2)-b(2k-3)(1), b(2k-2)(n) with offset 1 (which will contain only one term), is written, and then the sequence b1(1); b1(2),b2(1); b1(3),b2(2),b3(1); ...; b1(2k-2), b2(2k-3), b3(2k-4), ..., b(2k-2)(1) is obtained, then this sequence will be identical to the first 2k^2-3k+1 terms of a(n), except that the first term of this sequence will be negative, the next two terms will be positive, the next three will be negative, the next four positive, and so on.
%C A053218 Subtriangle of triangle in A152920. - _Philippe Deléham_, Nov 21 2011
%F A053218 T(n, k) = n*2^(k-1) - (k-1)*2^(k-2). - _Ya-Ping Lu_, Mar 24 2023
%e A053218 Triangle T(n,k) begins:
%e A053218 1;
%e A053218 2, 3;
%e A053218 3, 5, 8;
%e A053218 4, 7, 12, 20;
%e A053218 5, 9, 16, 28, 48;
%e A053218 6, 11, 20, 36, 64, 112;
%e A053218 7, 13, 24, 44, 80, 144, 256;
%e A053218 ...
%t A053218 NestList[FoldList[Plus, #[[1]] + 1, #] &, {1}, 10] // Grid (* _Geoffrey Critzer_, Jun 27 2013 *)
%Y A053218 Cf. A053219 (reverse of this triangle), A053220 (center elements), A053221 (row sums), A001792, A045618, A152920.
%K A053218 easy,nonn,tabl,changed
%O A053218 1,2
%A A053218 _Asher Auel_, Jan 01 2000