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%I #22 Jan 14 2018 14:51:45
%S 1,1,1,1,2,1,1,3,3,1,2,4,6,4,2,1,6,10,10,6,1,1,7,16,20,16,7,1,3,8,23,
%T 36,36,23,8,3,3,11,31,59,72,59,31,11,3,1,14,42,90,131,131,90,42,14,1,
%U 2,15,56,132,221,262,221,132,56,15,2,4,17,71,188,353,483,483,353,188,71,17,4,6,21,88,259,541,836,966,836,541,259
%N Array read by antidiagonals: Pascal-like recursion and self-referential boundaries.
%C Array with recursion T(i,j) = T(i-1,j) + T(i,j-1), and boundaries T(0,n) = T(n,0) = a(n). Here a(n) is the array T read by antidiagonals. Require that a(0)=a(1)=1.
%H Alex Meiburg, <a href="/A297359/b297359.txt">Table of n, a(n) for n = 1..19999</a>
%e The array looks like
%e 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, ...
%e 1, 2, 3, 4, 6, 7, 8, 11, 14, 15, 17, ...
%e 1, 3, 6, 10, 16, 23, 31, 42, 56, 71, 88, ...
%e 1, 4, 10, 20, 36, 59, 90, 132, 188, 259, 347, ...
%e 2, 6, 16, 36, 72, 131, 221, 353, 541, 800, ...
%e 1, 7, 23, 59, 131, 262, 483, 836, 1377, ...
%e 1, 8, 31, 90, 221, 483, 966, 1802, ...
%e 3, 11, 42, 132, 353, 836, 1802, ...
%e 3, 14, 56, 188, 541, 1377, ...
%e 1, 15, 71, 259, 800, ...
%e 2, 17, 88, 347, ...
%e ... [Table corrected and reformatted by _Jon E. Schoenfield_, Jan 14 2018]
%e 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 t[a_, b_] := (t[a, b] = t[a, b - 1] + t[a - 1, b]);
%t t[0, x_] := a[x]; t[x_, 0] := a[x];
%t a[0] = 1; a[1] = 1;
%t a[x_] := With[{k = Floor[(Sqrt[8 x + 1] - 1)/2]},
%t t[x - k (k + 1)/2, (k + 1) (k + 2)/2 - x - 1]]
%t a /@ Range[60]
%t TableForm[ Table[t[i, j], {i, 0, 5}, {j, 0, 12}]]
%Y Cf. A283682, A283681, A283683.
%Y See also A007318, A297495, A297497, A297188 (antidiagonal sums).
%K nonn,tabl,easy,nice
%O 1,5
%A _Alex Meiburg_, Dec 29 2017