A101866 Array read by antidiagonals: Arnoux's product T(n,k) = n * k = nk + ceiling(phi n) ceiling(phi k), where phi = (1 + sqrt(5))/2 ; m, n >= 1.

5, 10, 10, 13, 20, 13, 18, 26, 26, 18, 23, 36, 34, 36, 23, 26, 46, 47, 47, 46, 26, 31, 52, 60, 65, 60, 52, 31, 34, 62, 68, 83, 83, 68, 62, 34, 39, 68, 81, 94, 106, 94, 81, 68, 39, 44, 78, 89, 112, 120, 120, 112, 89, 78, 44, 47, 88, 102, 123, 143, 136, 143, 123, 102, 88, 47, 52

Offset: 1

Author

N. J. A. Sloane, Jan 28 2005

Keywords

• nonn
• tabl

Comments

Row 1 / column 1 (given in A101868) = positions of 1 in A188009, viz.,

A188009 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, ...), A101868 = (5, 10, 13, 18, 23, 26, 31, 34, 39, 44, 47, 52, 57, ...). - Clark Kimberling and John W. Layman, Mar 19 2011, corrected and edited by M. F. Hasler, Oct 12 2017

By definition, the array is symmetric, so row n = column n. Row 1 is essentially the same as A188434: T(n,1) = A101868(n) = A188434(n+1). - M. F. Hasler, Oct 12 2017

This product is commutative but is not associative and does not distribute over addition. - Peter Bala, Aug 13 2022

Examples

			   5 10 13 18  23 ...
  10 20 26 36  46
  13 26 34 47  60
  18 36 47 65  83
  23 46 60 83 106
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
• P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
• P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (No. 4, 1989), 319-320. [Annotated scanned copy]

Crossrefs

Cf. A101858, A101330, A101385, A101633 for similarly defined products.

Main diagonal is A101867.

First 3 rows are A101868, A101869, A101870.

Cf. A001622.

Programs

Mathematica

A101866[n_, k_] := n*k + Ceiling[n*GoldenRatio]*Ceiling[k*GoldenRatio];
Table[A101866[n-k+1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

PARI

T(n, k) = my(phi = (1+sqrt(5))/2); n*k + ceil(phi*n)*ceil(phi*k); \\ Michel Marcus, Mar 29 2016