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

Author

Alex Meiburg, Dec 29 2017

Keywords

• nonn
• tabl
• easy
• nice

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A283682, A283681, A283683.

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

Programs

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}]]