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%I #20 Oct 02 2022 16:01:30
%S 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,
%T 16,18,18,16,11,8,9,13,18,22,21,22,18,13,9,10,15,21,25,26,26,25,21,15,
%U 10,11,16,24,29,29,32,29,29,24,16,11,12,18,26,33,34,36,36,34,33,26,18,12
%N 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.
%C 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]
%C 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.
%H P. Grabner et al., <a href="http://dx.doi.org/10.1016/0893-9659(94)90017-5">Associativity of recurrence multiplication</a>, Appl. Math. Lett. 7 (1994), 85-90.
%H D. E. Knuth, <a href="http://dx.doi.org/10.1016/0893-9659(88)90176-0">Fibonacci multiplication</a>, Appl. Math. Lett. 1 (1988), 57-60.
%H W. F. Lunnon, <a href="/A101330/a101330.txt">Proof of formula</a>
%F T(n, k) = n*k - [(k+1)/phi^2] [(n+1)/phi^2]. For proof see link. - _Fred Lunnon_, May 24 2008
%e Array begins:
%e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...
%e 2 3 5 7 8 10 11 13 15 16 18 20 21 23 24 26 28 29 31 ...
%e 3 5 8 11 13 16 18 21 24 26 29 32 34 37 39 42 45 47 50 ...
%e 4 7 11 15 18 22 25 29 33 36 40 44 47 51 54 58 62 65 69 ...
%e 5 8 13 18 21 26 29 34 39 42 47 52 55 60 63 68 73 76 81 ...
%e ...
%t T[n_, k_] := With[{phi2 = GoldenRatio^2}, n k - Floor[(k + 1)/phi2] Floor[ (n + 1)/phi2]];
%t Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 31 2020 *)
%o (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
%Y Cf. A101330, A101385, A035517, A014417. Main diagonal is A101711.
%Y First 4 rows give A000027, A022342, A026274, A101741.
%K nonn,tabl,easy
%O 1,2
%A _David Applegate_ and _N. J. A. Sloane_, Jan 26 2005
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