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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.

%I A101866 #37 Mar 20 2024 11:49:45
%S A101866 5,10,10,13,20,13,18,26,26,18,23,36,34,36,23,26,46,47,47,46,26,31,52,
%T A101866 60,65,60,52,31,34,62,68,83,83,68,62,34,39,68,81,94,106,94,81,68,39,
%U A101866 44,78,89,112,120,120,112,89,78,44,47,88,102,123,143,136,143,123,102,88,47,52
%N 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.
%C A101866 Row 1 / column 1 (given in A101868) = positions of 1 in A188009, viz.,
%C A101866   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
%C A101866 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
%C A101866 This product is commutative but is not associative and does not distribute over addition. - _Peter Bala_, Aug 13 2022
%H A101866 Paolo Xausa, <a href="/A101866/b101866.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).
%H A101866 P. Arnoux, <a href="http://dx.doi.org/10.1016/0893-9659(89)90078-5">Some remarks about Fibonacci multiplication</a>, Appl. Math. Lett. 2 (1989), 319-320.
%H A101866 P. Arnoux, <a href="/A101858/a101858.pdf">Some remarks about Fibonacci multiplication</a>, Appl. Math. Lett. 2 (No. 4, 1989), 319-320. [Annotated scanned copy]
%e A101866    5 10 13 18  23 ...
%e A101866   10 20 26 36  46
%e A101866   13 26 34 47  60
%e A101866   18 36 47 65  83
%e A101866   23 46 60 83 106
%e A101866   ...
%t A101866 A101866[n_, k_] := n*k + Ceiling[n*GoldenRatio]*Ceiling[k*GoldenRatio];
%t A101866 Table[A101866[n-k+1, k], {n, 15}, {k, n}] (* _Paolo Xausa_, Mar 20 2024 *)
%o A101866 (PARI) T(n, k) = my(phi = (1+sqrt(5))/2); n*k + ceil(phi*n)*ceil(phi*k); \\ _Michel Marcus_, Mar 29 2016
%Y A101866 Cf. A101858, A101330, A101385, A101633 for similarly defined products.
%Y A101866 Main diagonal is A101867.
%Y A101866 First 3 rows are A101868, A101869, A101870.
%Y A101866 Cf. A001622.
%K A101866 nonn,tabl
%O A101866 1,1
%A A101866 _N. J. A. Sloane_, Jan 28 2005