|
%I #11 Jan 18 2020 03:12:34
%S 0,1,1,0,2,0,1,2,0,1,1,1,3,0,1,1,0,2,1,2,3,0,1,1,1,2,0,1,2,1,2,2,2,4,
%T 0,1,1,0,2,1,2,2,0,1,1,1,3,1,2,2,1,3,2,3,4,0,1,1,1,2,0,1,2,1,2,2,2,3,
%U 0,1,1,0,2,1,2,3,1,2,2,2,3,1,2,3,2,3,3
%N The number of common terms in the Zeckendorf and dual Zeckendorf representations of n.
%C The indices of records are numbers of the form F(2*k - 1) - 1, for k > 0, where F(k) is the k-th Fibonacci number. The corresponding record values are k - 1 = 0, 1, 2, ...
%H Amiram Eldar, <a href="/A331466/b331466.txt">Table of n, a(n) for n = 0..10000</a>
%F a(A000045(2*n - 1) - 1) = a(A000045(2*n) - 1) = n - 1.
%F a(A000045(n)) = a(A331467(n)) = 0 for n > 2.
%e a(6) = 1 since the Zeckendorf representation of 6 is 1001 (i.e., F(2) + F(5)), its dual Zeckendorf representation is 111 (i.e., F(2) + F(3) + F(4)), and there is only one position with a common digit 1, corresponding to the one common summand F(2).
%t m = 1000; zeck = Select[Range[0, m], BitAnd[#, 2 #] == 0 &]; dualZeck = Select[Range[0, m], SequenceCount[IntegerDigits[#, 2], {0, 0}] == 0 &]; DigitCount[BitAnd[zeck[[#]], dualZeck[[#]]] & /@ Range[Min[Length[zeck], Length[dualZeck]]], 2, 1]
%Y Cf. A000045, A007895, A014417, A104326, A112310, A331467.
%K nonn,base
%O 0,5
%A _Amiram Eldar_, Jan 17 2020
|
