A101870 -id:A101870 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 28 2005

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

			   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
  ...

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]

Cf. A101858, A101330, A101385, A101633 for similarly defined products.
Main diagonal is A101867.
First 3 rows are A101868, A101869, A101870.
Cf. A001622.

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 *)

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

A188471 Positions of 0 in A188470.

Original entry on oeis.org

5, 13, 26, 34, 47, 60, 68, 81, 89, 102, 115, 123, 136, 149, 157, 170, 178, 191, 204, 212, 225, 233, 246, 259, 267, 280, 293, 301, 314, 322, 335, 348, 356, 369, 382, 390, 403, 411, 424, 437, 445, 458, 466, 479, 492, 500, 513, 526, 534, 547, 555, 568, 581, 589, 602, 610, 623, 636, 644, 657, 670, 678, 691, 699, 712, 725, 733, 746, 759, 767, 780, 788

Offset: 1

Clark Kimberling, Apr 01 2011

nonn

See A188294, A188470.

Same as A101870 with 5 prepended. This follows from A188470 = 1 - A187946 except for n=5, cf. FORMULA there. - M. F. Hasler, Oct 12 2017

Cf. A188294, A188470.

Mathematica
```
(See A188471.)
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cp's OEIS Frontend

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.

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