A135090 Array read by antidiagonals: T(n, k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 0, k >= 0.

0, 0, 0, 0, 3, 0, 0, 5, 5, 0, 0, 8, 8, 8, 0, 0, 11, 13, 13, 11, 0, 0, 13, 18, 21, 18, 13, 0, 0, 16, 21, 29, 29, 21, 16, 0, 0, 18, 26, 34, 40, 34, 26, 18, 0, 0, 21, 29, 42, 47, 47, 42, 29, 21, 0, 0, 24, 34, 47, 58, 55, 58, 47, 34, 24, 0, 0, 26, 39, 55, 65, 68, 68, 65, 55, 39, 26, 0, 0, 29, 42, 63, 76, 76, 84, 76, 76, 63, 42, 29, 0, 0, 32, 47

Offset

0,5

Comments

This is a variant of A101330. See that entry for much more information.

Links

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Formula

T(n, k) = 3*n*k - n*h(k) - k*h(n) where h(n) = A060144(n + 1). - Peter Luschny, Mar 21 2024

Example

Array begins:
  n\k |  0   1   2   3   4    5    6    7    8    9 ...
  ----+------------------------------------------------
   0  |  0   0   0   0   0    0    0    0    0    0 ...
   1  |  0   3   5   8  11   13   16   18   21   24 ...
   2  |  0   5   8  13  18   21   26   29   34   39 ...
   3  |  0   8  13  21  29   34   42   47   55   63 ...
   4  |  0  11  18  29  40   47   58   65   76   87 ...
   5  |  0  13  21  34  47   55   68   76   89  102 ...
   6  |  0  16  26  42  58   68   84   94  110  126 ...
   7  |  0  18  29  47  65   76   94  105  123  141 ...
   8  |  0  21  34  55  76   89  110  123  144  165 ...
   9  |  0  24  39  63  87  102  126  141  165  189 ...
  ...

Maple

h := n -> floor(2*(n + 1)/(sqrt(5) + 3)):  # A060144(n+1)
T := (n, k) -> 3*n*k - n*h(k) - k*h(n):
seq(print(seq(T(n, k), k = 0..9)), n = 0..7);  # Peter Luschny, Mar 21 2024

Mathematica

A135090[n_, k_] := 3*n*k - n*Floor[(k + 1) / GoldenRatio^2] - k*Floor[(n + 1) / GoldenRatio^2];
Table[A135090[n-k, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Mar 21 2024 *)

Crossrefs

Cf. A101330, A060144, A001622.

Sequence in context: A362271 A115013 A072736 * A303690 A304156 A305509

Adjacent sequences: A135087 A135088 A135089 * A135091 A135092 A135093

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 17 2008

Status

approved