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

1,1

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A349462 A340761 A035625 * A132128 A280057 A257174

Adjacent sequences: A219872 A219873 A219874 * A219876 A219877 A219878

Keyword

nonn,tabl

Author

Michel Marcus, Dec 01 2012

Status

approved