%I #18 Mar 21 2024 08:34:01

%S 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 58,51,42,19,32,58,63,76,74,58,50,21,37,64,76,87,97,84,69,55,24,42,74,

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

%C This is a hybrid of the Porta-Stolarsky star product (A101858) and the Arnoux product (A101866)

%H Paolo Xausa, <a href="/A295573/b295573.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).

%H 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 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 The array begins:

%e 3, 6, 8, 11, 14, 16, 19, 21, 24, 27, 29, 32, ...

%e 8, 16, 21, 29, 37, 42, 50, 55, 63, 71, 76, 84, ...

%e 11, 22, 29, 40, 51, 58, 69, 76, 87, 98, 105, 116, ...

%e 16, 32, 42, 58, 74, 84, 100, 110, 126, 142, 152, 168, ...

%e 21, 42, 55, 76, 97, 110, 131, 144, 165, 186, 199, 220, ...

%e 24, 48, 63, 87, 111, 126, 150, 165, 189, 213, 228, 252, ...

%e 29, 58, 76, 105, 134, 152, 181, 199, 228, 257, 275, 304, ...

%e 32, 64, 84, 116, 148, 168, 200, 220, 252, 284, 304, 336, ...

%e ...

%p T := proc(n, k) local phi;

%p phi := (1+sqrt(5))/2 ;

%p n*k+floor(n*phi)*ceil(phi*k) ;

%p end proc:

%p for n from 1 to 12 do

%p lprint([seq(T(n-i+1,i),i=1..n)]);

%p od: # by antidiagonals

%p for n from 1 to 12 do

%p lprint([seq(T(n,i),i=1..12)]);

%p od: # by rows

%t A295573[n_, k_] := n*k + Floor[n * GoldenRatio] * Ceiling[k * GoldenRatio];

%t Table[A295573[n-k+1,k], {n, 15}, {k, n}] (* _Paolo Xausa_, Mar 20 2024 *)

%Y Cf. A001622, A101858, A101866, A371382 (main diagonal).

%K nonn,tabl

%O 1,1

%A _N. J. A. Sloane_, Dec 03 2017