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A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

%I A361989 #30 Mar 21 2024 08:37:08
%S A361989 0,0,1,0,2,1,0,4,3,2,1,0,7,6,5,4,3,2,1,0,12,11,10,9,8,7,6,5,4,3,2,1,0,
%T A361989 20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,33,32,31,30,29,
%U A361989 28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0
%N A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).
%C A361989 We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.
%C A361989 The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
%C A361989 This sequence can also be seen as an irregular table T(n, k), n > 0, k = 1..A000045(n), where T(n, k) = A000045(n) - k.
%C A361989 a(n-1) for n>=1 is the starting position of the first occurrence of one of the longest words w in the Fibonacci word A003849 such that no length-n factor of w is repeated. The length of such words is 2n. (See links) - _Gandhar Joshi_, Mar 19 2024
%H A361989 Gandhar Joshi, <a href="/A361989/a361989_2.txt">Walnut code and details</a>
%H A361989 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>
%F A361989 a(n) = A000045(A072649(n)) - A194029(n) for n > 0.
%F A361989 a(n) = A130312(n) - A194029(n) for n > 0.
%e A361989 For n = 42:
%e A361989 - using F(k) = A000045(k),
%e A361989 - the dual Zeckendorf representation of 42 is F(8) + F(7) + F(5) + F(3) + F(2),
%e A361989 - the numbers F(6) and F(4) are missing,
%e A361989 - so a(42) = F(6) + F(4) = 8 + 3 = 11.
%e A361989 .
%e A361989 As an irregular triangle the sequence begins:
%e A361989      0;
%e A361989      0;
%e A361989      1,  0;
%e A361989      2,  1,  0;
%e A361989      4,  3,  2, 1, 0;
%e A361989      7,  6,  5, 4, 3, 2, 1, 0;
%e A361989     12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0;
%e A361989     ...
%o A361989 (PARI) for (n = 1, 9, for (k = 1, f = fibonacci(n), print1 (f-k", ")))
%Y A361989 Cf. A000045, A003754, A022290, A035327, A072649, A130312, A132665, A194029, A356771.
%K A361989 nonn,base,tabf
%O A361989 0,5
%A A361989 _Rémy Sigrist_, Apr 02 2023