A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1

Offset

0,3

Comments

The row sums equal A052952.

Let the triangle = M. Then lim_{n->infinity} M^n = A173285 as a left-shifted vector.

A173284 * [1, 2, 3, ...] = A054451: (1, 1, 4, 5, 12, 17, 33, ...). - Gary W. Adamson, Mar 03 2010

From Johannes W. Meijer, Sep 05 2013: (Start)

Triangle read by rows formed from antidiagonals of triangle A104762.

The diagonal sums lead to A004695. (end)

Links

Table of n, a(n) for n=0..62.

Formula

Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

From Johannes W. Meijer, Sep 05 2013: (Start)

T(n,k) = A000045(n-2*k+1), n >= 0 and 0 <= k <= floor(n/2).

T(n,k) = A104762(n-k, k). (End)

Example

First few rows of the triangle:
    1;
    1;
    2,   1;
    3,   1;
    5,   2,  1;
    8,   3,  1;
   13,   5,  2,  1;
   21,   8,  3,  1;
   34,  13,  5,  2,  1;
   55,  21,  8,  3,  1;
   89,  34, 13,  5,  2, 1;
  144,  55, 21,  8,  3, 1;
  233,  89, 34, 13,  5, 2, 1;
  377, 144, 55, 21,  8, 3, 1;
  610, 233, 89, 34, 13, 5, 2, 1;
  ...

Maple

T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013

Crossrefs

Cf. A000045, A004695, A052952, A054451, A173285.

Cf. (Similar triangles) A008315 (Catalan), A011973 (Pascal), A102541 (Losanitsch), A122196 (Fractal), A122197 (Fractal), A128099 (Pell-Jacobsthal), A152198, A152204, A207538, A209634.

Sequence in context: A147486 A168018 A173238 * A278136 A085053 A296118

Adjacent sequences: A173281 A173282 A173283 * A173285 A173286 A173287

Keyword

nonn,tabf

Author

Gary W. Adamson, Feb 14 2010

Extensions

Term a(15) corrected by Johannes W. Meijer, Sep 05 2013

Status

approved