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A356969 A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.

%I A356969 #11 Sep 10 2022 14:00:53
%S A356969 0,0,0,0,1,0,0,0,0,0,0,1,2,1,0,0,1,2,2,1,0,0,0,0,3,0,0,0,0,1,2,1,1,2,
%T A356969 1,0,0,0,2,2,4,2,2,0,0,0,1,2,3,3,3,3,2,1,0,0,1,2,2,4,5,4,2,2,1,0,0,0,
%U A356969 0,3,0,5,5,0,3,0,0,0,0,1,2,1,1,2,6,2,1,1,2,1,0
%N A356969 A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.
%C A356969 The dual Zeckendorf representation corresponds to the lazy Fibonacci representation.
%C A356969 See A334348 for the sequence dealing with Zeckendorf (or greedy Fibonacci) representations. Unlike A334348, the present sequence is not associative.
%H A356969 Rémy Sigrist, <a href="/A356969/a356969.png">Colored representation of the table for x, y < Fibonacci(16)-1</a> (the hue is function of A(x,y))
%H A356969 Rémy Sigrist, <a href="/A356969/a356969.gp.txt">PARI program</a>
%H A356969 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>
%F A356969 A(n, k) = A022290(A003754(n+1) AND A003754(k+1)) (where AND denotes the bitwise AND operator, A004198).
%F A356969 A(n, k) = A(k, n).
%F A356969 A(n, 0) = 0.
%F A356969 A(n, n) = n.
%e A356969 Square array A(n, k) begins:
%e A356969   n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12  13
%e A356969   ---+----------------------------------------------
%e A356969     0|  0  0  0  0  0  0  0  0  0  0   0   0   0   0
%e A356969     1|  0  1  0  1  1  0  1  0  1  1   0   1   1   0
%e A356969     2|  0  0  2  2  0  2  2  2  2  0   2   2   0   2
%e A356969     3|  0  1  2  3  1  2  3  2  3  1   2   3   1   2
%e A356969     4|  0  1  0  1  4  3  4  0  1  4   3   4   4   3
%e A356969     5|  0  0  2  2  3  5  5  2  2  3   5   5   3   5
%e A356969     6|  0  1  2  3  4  5  6  2  3  4   5   6   4   5
%e A356969     7|  0  0  2  2  0  2  2  7  7  5   7   7   0   2
%e A356969     8|  0  1  2  3  1  2  3  7  8  6   7   8   1   2
%e A356969     9|  0  1  0  1  4  3  4  5  6  9   8   9   4   3
%e A356969    10|  0  0  2  2  3  5  5  7  7  8  10  10   3   5
%e A356969    11|  0  1  2  3  4  5  6  7  8  9  10  11   4   5
%e A356969    12|  0  1  0  1  4  3  4  0  1  4   3   4  12  11
%e A356969    13|  0  0  2  2  3  5  5  2  2  3   5   5  11  13
%o A356969 (PARI) See Links section.
%Y A356969 Cf. A003754, A022290, A334348.
%K A356969 nonn,base,tabl
%O A356969 0,13
%A A356969 _Rémy Sigrist_, Sep 06 2022