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A295573 Array read by upwards antidiagonals: T(n,k) = nk + floor(phi n) ceiling(phi k) where phi = (1 + sqrt(5))/2.

%I A295573 #18 Mar 21 2024 08:34:01
%S A295573 3,8,6,11,16,8,16,22,21,11,21,32,29,29,14,24,42,42,40,37,16,29,48,55,
%T A295573 58,51,42,19,32,58,63,76,74,58,50,21,37,64,76,87,97,84,69,55,24,42,74,
%U A295573 84,105,111,110,100,76,63,27,45,84,97,116,134,126,131,110,87,71,29,50,90,110,134,148,152,150,144,126,98,76,32
%N A295573 Array read by upwards antidiagonals: T(n,k) =  nk + floor(phi n) ceiling(phi k) where phi = (1 + sqrt(5))/2.
%C A295573 This is a hybrid of the Porta-Stolarsky star product (A101858) and the Arnoux product (A101866)
%H A295573 Paolo Xausa, <a href="/A295573/b295573.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).
%H A295573 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 (No. 4, 1989), 319-320.
%H A295573 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 A295573 The array begins:
%e A295573 3, 6, 8, 11, 14, 16, 19, 21, 24, 27, 29, 32, ...
%e A295573 8, 16, 21, 29, 37, 42, 50, 55, 63, 71, 76, 84, ...
%e A295573 11, 22, 29, 40, 51, 58, 69, 76, 87, 98, 105, 116, ...
%e A295573 16, 32, 42, 58, 74, 84, 100, 110, 126, 142, 152, 168, ...
%e A295573 21, 42, 55, 76, 97, 110, 131, 144, 165, 186, 199, 220, ...
%e A295573 24, 48, 63, 87, 111, 126, 150, 165, 189, 213, 228, 252, ...
%e A295573 29, 58, 76, 105, 134, 152, 181, 199, 228, 257, 275, 304, ...
%e A295573 32, 64, 84, 116, 148, 168, 200, 220, 252, 284, 304, 336, ...
%e A295573 ...
%p A295573 T := proc(n, k) local phi;
%p A295573         phi := (1+sqrt(5))/2 ;
%p A295573         n*k+floor(n*phi)*ceil(phi*k) ;
%p A295573 end proc:
%p A295573 for n from 1 to 12 do
%p A295573 lprint([seq(T(n-i+1,i),i=1..n)]);
%p A295573 od: # by antidiagonals
%p A295573 for n from 1 to 12 do
%p A295573 lprint([seq(T(n,i),i=1..12)]);
%p A295573 od: # by rows
%t A295573 A295573[n_, k_] := n*k + Floor[n * GoldenRatio] * Ceiling[k * GoldenRatio];
%t A295573 Table[A295573[n-k+1,k], {n, 15}, {k, n}] (* _Paolo Xausa_, Mar 20 2024 *)
%Y A295573 Cf. A001622, A101858, A101866, A371382 (main diagonal).
%K A295573 nonn,tabl
%O A295573 1,1
%A A295573 _N. J. A. Sloane_, Dec 03 2017