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

Author

Amiram Eldar, Jan 17 2020

Keywords

• nonn
• base

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.

Examples

			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).
		

Links

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

Crossrefs

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

Programs

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

Formula

A331466(a(n)) = 0.