A331467 Numbers with no common terms in their Zeckendorf and dual Zeckendorf representations.

0, 3, 5, 8, 13, 16, 21, 26, 34, 37, 42, 55, 60, 68, 71, 89, 92, 97, 110, 115, 144, 149, 157, 160, 178, 181, 186, 233, 236, 241, 254, 259, 288, 293, 301, 304, 377, 382, 390, 393, 411, 414, 419, 466, 469, 474, 487, 492, 610, 613, 618, 631, 636, 665, 670, 678, 681

Offset

1,2

Comments

Include all the Fibonacci numbers > 2.

The number of terms <= F(k), the k-th Fibonacci number, is A000931(k + 5), for k > 3.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Formula

A331466(a(n)) = 0.

Example

3 is a term since its Zeckendorf representation is 100 (i.e., F(4)), its dual Zeckendorf representation is 11 (i.e., F(2) + F(3)), and there is no position with the digit 1 common to both representations (i.e., the Fibonacci summands are different).

Mathematica

m = 10^4; zeck = Select[Range[0, m], BitAnd[#, 2 #] == 0 &]; dualZeck = Select[Range[0, m], SequenceCount[IntegerDigits[#, 2], {0, 0}] == 0 &]; s = DigitCount[BitAnd[zeck[[#]], dualZeck[[#]]] & /@ Range[Min[Length[zeck], Length[dualZeck]]], 2, 1]; -1 + Position[s, _?(# == 0 &)] // Flatten

Crossrefs

Cf. A000045, A000931, A007895, A014417, A104326, A112310, A331466.

Sequence in context: A129141 A289013 A357048 * A337289 A097431 A123929

Adjacent sequences: A331464 A331465 A331466 * A331468 A331469 A331470

Keyword

nonn,base

Author

Amiram Eldar, Jan 17 2020

Status

approved