A219875 Multiplication table of the operation "n o m" = n*m + ceiling(n/phi)* ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

2, 4, 4, 5, 8, 5, 7, 10, 10, 7, 9, 14, 13, 14, 9, 10, 18, 18, 18, 18, 10, 12, 20, 23, 25, 23, 20, 12, 13, 24, 26, 32, 32, 26, 24, 13, 15, 26, 31, 36, 41, 36, 31, 26, 15, 17, 30, 34, 43, 46, 46, 43, 34, 30, 17, 18, 34, 39, 47, 55, 52, 55, 47, 39, 34, 18

Offset: 1

Author

Michel Marcus, Dec 01 2012

Keywords

• nonn
• tabl

Comments

Like A101866, this operation is associative.

First rows of the table are:

1: 2, 4, 5, 7, 9, 10, 12, 13, 15, 17, ...

2: 4, 8, 10, 14, 18, 20, 24, 26, 30, 34, ...

3: 5, 10, 13, 18, 23, 26, 31, 34, 39, 44, ...

4: 7, 14, 18, 25, 32, 36, 43, 47, 54, 61, ...

5: 9, 18, 23, 32, 41, 46, 55, 60, 69, 78, ...

6:10, 20, 26, 36, 46, 52, 62, 68, 78, 88, ...

7:12, 24, 31, 43, 55, 62, 74, 81, 93, 105, ...

8:13, 26, 34, 47, 60, 68, 81, 89, 102, 115, ...

9:15, 30, 39, 54, 69, 78, 93, 102, 117, 132, ...

Row 1 is A004956.

Row 3 is A101868.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
• P. Arnoux, Some remarks about Fibonacci multiplication, Applied Mathematics Letters, Volume 2, Issue 4, 1989, Pages 319-320.

Crossrefs

Cf. A001622, A004956, A101385, A101858, A101866, A101868, A371381 (main diagonal).

Programs

Mathematica

A219875[n_, m_] := n*m + Ceiling[n / GoldenRatio] * Ceiling[m / GoldenRatio];
Table[A219875[n-m+1, m], {n, 15}, {m, n}] (* Paolo Xausa, Mar 20 2024 *)

PARI

prod(m,n) = {phi = (1+sqrt(5))/2; return (m*n + ceil(m/phi)*ceil(n/phi));}