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.
Triangle begins: 0, 1,1, 1,2,3, 1,2,4,7, 2,3,5,9,16, 3,5,8,13,22,38, 1,4,9,17,30,52,90, 2,3,7,16,33,63,115,205, 4,6,9,16,32,65,128,243,448, 7,11,17,26,42,74,139,267,510,958, 2,9,20,37,63,105,179,318,585,1095,2053, 3,5,14,34,71,134,239,418,736,1321,2416,4469, 5,8,13,27,61,132,266,505,923,1659,2980,5396,9865, 9,14,22,35,62,123,255,521,1026,1949,3608,6588,11984,21849, 16,25,39,61,96,158,281,536,1057,2083,4032,7640,14228,26212,48061, ...
The array looks like 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, ... 1, 2, 3, 4, 6, 7, 8, 11, 14, 15, 17, ... 1, 3, 6, 10, 16, 23, 31, 42, 56, 71, 88, ... 1, 4, 10, 20, 36, 59, 90, 132, 188, 259, 347, ... 2, 6, 16, 36, 72, 131, 221, 353, 541, 800, ... 1, 7, 23, 59, 131, 262, 483, 836, 1377, ... 1, 8, 31, 90, 221, 483, 966, 1802, ... 3, 11, 42, 132, 353, 836, 1802, ... 3, 14, 56, 188, 541, 1377, ... 1, 15, 71, 259, 800, ... 2, 17, 88, 347, ... ... [Table corrected and reformatted by _Jon E. Schoenfield_, Jan 14 2018] The defining property is that when this array is read by antidiagonals we get 1,1,1,1,2,1,... which is both the sequence itself and the top row and first column of the array.
t[a_, b_] := (t[a, b] = t[a, b - 1] + t[a - 1, b]);
t[0, x_] := a[x]; t[x_, 0] := a[x];
a[0] = 1; a[1] = 1;
a[x_] := With[{k = Floor[(Sqrt[8 x + 1] - 1)/2]},
t[x - k (k + 1)/2, (k + 1) (k + 2)/2 - x - 1]]
a /@ Range[60]
TableForm[ Table[t[i, j], {i, 0, 5}, {j, 0, 12}]]
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