A108285 Triangle read by rows, generated from (1, 2, 3, ...).

1, 1, 3, 1, 4, 6, 1, 5, 11, 10, 1, 6, 18, 26, 15, 1, 7, 27, 58, 57, 21, 1, 8, 38, 112, 179, 120, 28, 1, 9, 51, 194, 453, 543, 247, 36

Offset

0,3

Comments

By diagonals (d=1,2,3,...) going to the left with (1,3,6,...) = d(1), these are sequences of the form (k-th term a(k) = d*a(k-1) + k). Example: 1, 7, 38, 194, ... (the 5th diagonal) = A014827, is generated by a(k) = 5*a(k-1) + k. Diagonal 2 = (1, 4, 11, 26, ...) = A000295; Diagonal 3 = (1, 5, 18, ...) = A000340; Diagonal 4 = (1, 6, 27, ...) = A014825.

Triangle A108243 is generated by analogous operations from (..., 3, 2, 1) instead of (1, 2, 3, ...).

Links

Table of n, a(n) for n=0..35.

Formula

n-th column = f(x), x = 1, 2, 3, ...; x^(n) + 2*x^(n-1) + 3*x^(n-2) + ... + (n+1).

Example

4th column (offset) = 10, 26, 58, 112, ...= f(x), x = 1, 2, 3; x^3 + 2x^2 + 3x + 4.
First few rows of the triangle are:
  1;
  1, 3;
  1, 4, 6;
  1, 5, 11, 10;
  1, 6, 18, 26, 15;
  1, 7, 27, 58, 57, 21;
  1, 8, 38, 112, 179, 120, 28;
  ...

Crossrefs

Cf. A108286, A000295, A000349, A014825, A000217, A108283, A108284, A108286.

Sequence in context: A345229 A080851 A209518 * A207619 A209694 A286951

Adjacent sequences: A108282 A108283 A108284 * A108286 A108287 A108288

Keyword

nonn,tabl

Author

Gary W. Adamson, May 30 2005

Status

approved