

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.


4



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
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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+j2} (= 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), 8590.
D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 5760.
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 ...
...


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 (* JeanFrançois 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 (presumably!), A101741.
Sequence in context: A135646 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



