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A297359
Array read by antidiagonals: Pascal-like recursion and self-referential boundaries.
4
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, 36, 36, 23, 8, 3, 3, 11, 31, 59, 72, 59, 31, 11, 3, 1, 14, 42, 90, 131, 131, 90, 42, 14, 1, 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
OFFSET
1,5
COMMENTS
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.
LINKS
EXAMPLE
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.
MATHEMATICA
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}]]
CROSSREFS
See also A007318, A297495, A297497, A297188 (antidiagonal sums).
Sequence in context: A114162 A259074 A162981 * A338291 A029264 A215064
KEYWORD
nonn,tabl,easy,nice
AUTHOR
Alex Meiburg, Dec 29 2017
STATUS
approved