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A219875 Multiplication table of the operation "n o m" = n*m + ceiling(n/phi)* ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

%I A219875 #12 Mar 21 2024 07:00:01
%S A219875 2,4,4,5,8,5,7,10,10,7,9,14,13,14,9,10,18,18,18,18,10,12,20,23,25,23,
%T A219875 20,12,13,24,26,32,32,26,24,13,15,26,31,36,41,36,31,26,15,17,30,34,43,
%U A219875 46,46,43,34,30,17,18,34,39,47,55,52,55,47,39,34,18
%N A219875 Multiplication table of the operation "n o m" = n*m + ceiling(n/phi)* ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.
%C A219875 Like A101866, this operation is associative.
%C A219875 First rows of the table are:
%C A219875 1: 2,  4,  5,  7,  9, 10, 12,  13,  15,  17, ...
%C A219875 2: 4,  8, 10, 14, 18, 20, 24,  26,  30,  34, ...
%C A219875 3: 5, 10, 13, 18, 23, 26, 31,  34,  39,  44, ...
%C A219875 4: 7, 14, 18, 25, 32, 36, 43,  47,  54,  61, ...
%C A219875 5: 9, 18, 23, 32, 41, 46, 55,  60,  69,  78, ...
%C A219875 6:10, 20, 26, 36, 46, 52, 62,  68,  78,  88, ...
%C A219875 7:12, 24, 31, 43, 55, 62, 74,  81,  93, 105, ...
%C A219875 8:13, 26, 34, 47, 60, 68, 81,  89, 102, 115, ...
%C A219875 9:15, 30, 39, 54, 69, 78, 93, 102, 117, 132, ...
%C A219875 Row 1 is A004956.
%C A219875 Row 3 is A101868.
%H A219875 Paolo Xausa, <a href="/A219875/b219875.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).
%H A219875 P. Arnoux, <a href="http://dx.doi.org/10.1016/0893-9659(89)90078-5">Some remarks about Fibonacci multiplication</a>, Applied Mathematics Letters, Volume 2, Issue 4, 1989, Pages 319-320.
%t A219875 A219875[n_, m_] := n*m + Ceiling[n / GoldenRatio] * Ceiling[m / GoldenRatio];
%t A219875 Table[A219875[n-m+1, m], {n, 15}, {m, n}] (* _Paolo Xausa_, Mar 20 2024 *)
%o A219875 (PARI) prod(m,n) = {phi = (1+sqrt(5))/2; return (m*n + ceil(m/phi)*ceil(n/phi));}
%Y A219875 Cf. A001622, A004956, A101385, A101858, A101866, A101868, A371381 (main diagonal).
%K A219875 nonn,tabl
%O A219875 1,1
%A A219875 _Michel Marcus_, Dec 01 2012