A101646 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330). Sometimes called the "arroba" product.

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 5, 6, 8, 11, 11, 8, 6, 7, 10, 13, 15, 13, 10, 7, 8, 11, 16, 18, 18, 16, 11, 8, 9, 13, 18, 22, 21, 22, 18, 13, 9, 10, 15, 21, 25, 26, 26, 25, 21, 15, 10, 11, 16, 24, 29, 29, 32, 29, 29, 24, 16, 11, 12, 18, 26, 33, 34, 36, 36, 34, 33, 26, 18, 12

Offset

1,2

Comments

Let n = Sum_{i >= 2} eps(i) Fib_i and k = Sum_{j >= 2} eps(j) Fib_j be the Zeckendorf expansions of n and k, respectively (cf. A035517, A014417). The product of n and k is defined here to be n x k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j-2} (= T(n,k)). [Comment corrected by R. J. Mathar, Aug 07 2007]

Although now 1 is the multiplicative identity, in contrast to A101330, this multiplication is not associative. For example, as pointed out by Grabner et al., we have (4 x 7 ) x 9 = 25 x 9 = 198 but 4 x (7 x 9 ) = 4 x 54 = 195.

Links

Table of n, a(n) for n=1..78.

P. Grabner et al., Associativity of recurrence multiplication, Appl. Math. Lett. 7 (1994), 85-90.

D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.

W. F. Lunnon, Proof of formula

Formula

T(n, k) = n*k - [(k+1)/phi^2] [(n+1)/phi^2]. For proof see link. - Fred Lunnon, May 24 2008

Example

Array begins:
  1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 ...
  2 3  5  7  8 10 11 13 15 16 18 20 21 23 24 26 28 29 31 ...
  3 5  8 11 13 16 18 21 24 26 29 32 34 37 39 42 45 47 50 ...
  4 7 11 15 18 22 25 29 33 36 40 44 47 51 54 58 62 65 69 ...
  5 8 13 18 21 26 29 34 39 42 47 52 55 60 63 68 73 76 81 ...
...

Mathematica

T[n_, k_] := With[{phi2 = GoldenRatio^2}, n k - Floor[(k + 1)/phi2] Floor[ (n + 1)/phi2]];
Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2020 *)

Prog

(PARI) T(n, k) = my(phi2 = ((1+sqrt(5))/2)^2); n*k - floor((k+1)/phi2)*floor((n+1)/phi2); \\ Michel Marcus, Mar 29 2016

Crossrefs

Cf. A101330, A101385, A035517, A014417. Main diagonal is A101711.

First 4 rows give A000027, A022342, A026274, A101741.

Sequence in context: A360920 A334922 A301851 * A166079 A269381 A080677

Adjacent sequences: A101643 A101644 A101645 * A101647 A101648 A101649

Keyword

nonn,tabl,easy

Author

David Applegate and N. J. A. Sloane, Jan 26 2005

Status

approved