A331466 The number of common terms in the Zeckendorf and dual Zeckendorf representations of n.

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, 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, 0, 1, 1, 0, 2, 1, 2, 3, 1, 2, 2, 2, 3, 1, 2, 3, 2, 3, 3

Offset

0,5

Comments

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, ...

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

a(A000045(2*n - 1) - 1) = a(A000045(2*n) - 1) = n - 1.

a(A000045(n)) = a(A331467(n)) = 0 for n > 2.

Example

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).

Mathematica

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]

Crossrefs

Cf. A000045, A007895, A014417, A104326, A112310, A331467.

Sequence in context: A294168 A299196 A227698 * A166124 A134979 A112248

Adjacent sequences: A331463 A331464 A331465 * A331467 A331468 A331469

Keyword

nonn,base

Author

Amiram Eldar, Jan 17 2020

Status

approved