A297359 Array read by antidiagonals: Pascal-like recursion and self-referential boundaries.

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

Alex Meiburg, Table of n, a(n) for n = 1..19999

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

Cf. A283682, A283681, A283683.

See also A007318, A297495, A297497, A297188 (antidiagonal sums).

Sequence in context: A114162 A259074 A162981 * A338291 A029264 A215064

Adjacent sequences: A297356 A297357 A297358 * A297360 A297361 A297362

Keyword

nonn,tabl,easy,nice

Author

Alex Meiburg, Dec 29 2017

Status

approved