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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 ...

...

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-Fran├žois 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

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Last modified July 25 12:15 EDT 2021. Contains 346290 sequences. (Running on oeis4.)