A289207 a(n) = max(0, n-2).

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Jun 28 2017

Keywords

• nonn
• less

Comments

This simple sequence is such that there is one and only one array of differences D(n,k) where the first and the second upper subdiagonal is a(n).

The rows of this array are existing sequences of the OEIS, prepended with zeros:

row 0 is A118425,

row 1 is A006478,

row 2 is A001629,

row 3 is A010049,

row 4 is A006367,

row 5 is not in the OEIS.

It can be observed that a(n) is an autosequence of the first kind whose second kind mate is A199969. In addition, the structure of the array D(n,k) shows that the first row is an autosequence.

For n = 1 to 8, rows with only one leading zero are also autosequences.

Examples

			Array of differences begin:
   0,   0,   0,   0,  0,   0,  0,  1,  4, 12, 30, 68, ...
   0,   0,   0,   0,  0,   0,  1,  3,  8, 18, 38, 76, ...
   0,   0,   0,   0,  0,   1,  2,  5, 10, 20, 38, 71, ...
   0,   0,   0,   0,  1,   1,  3,  5, 10, 18, 33, 59, ...
   0,   0,   0,   1,  0,   2,  2,  5,  8, 15, 26, 46, ...
   0,   0,   1,  -1,  2,   0,  3,  3,  7, 11, 20, 34, ...
   0,   1,  -2,   3, -2,   3,  0,  4,  4,  9, 14, 24, ...
   1,  -3,   5,  -5,  5,  -3,  4,  0,  5,  5, 10, 16, ...
  -4,   8, -10,  10, -8,   7, -4,  5,  0,  6,  6, 17, ...
  12, -18,  20, -18, 15, -11,  9, -5,  6,  0,  7,  7, ...
  ...
		

Links

• OEIS Wiki, Autosequence
• Index entries for linear recurrences with constant coefficients, signature (2, -1).

Crossrefs

Essentially the same as A023444. Cf. A001477, A118425, A006478, A001629, A010049, A006367, A199969.

Programs

Mathematica

a[n_] := Max[0, n - 2];
D[n_, k_] /; k == n + 1 := a[n]; D[n_, k_] /; k == n + 2 := a[n]; D[n_, k_] /; k > n + 2 := D[n, k] = Sum[D[n + 1, j], {j, 0, k - 1}]; D[n_, k_] /; k <= n := D[n, k] = D[n - 1, k + 1] - D[n - 1, k];
Table[D[n, k], {n, 0, 11}, {k, 0, 11}]

Formula

G.f.: x^3 / (1-x)^2.