A063732 Numbers whose Lucas representation excludes L_0 = 2.

0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90

Offset

1,3

Comments

From Michel Dekking, Aug 26 2019: (Start)

This sequence is a generalized Beatty sequence. We know that A054770, the sequence of numbers whose Lucas representation includes L_0=2, is equal to A054770(n) = A000201(n) + 2*n - 1 = floor((phi+2)*n) - 1.

One also easily checks that the numbers 3-phi and phi+2 form a Beatty pair. This implies that the sequence with terms floor((3-phi)*n)-1 is the complement of A054770 in the natural numbers 0,1,2,...

It follows that a(n) = 3*n - floor(n*phi) - 2.

(End)

Links

Table of n, a(n) for n=1..66.

Formula

a(n) = floor((3-phi)*n)-1, where phi is the golden mean. - Michel Dekking, Aug 26 2019

Crossrefs

Cf. A003622, A022342. Complement of A054770.

Partial sums of A003842.

Cf. A130310 (Lucas representation).

Sequence in context: A298108 A330993 A184744 * A047367 A184620 A039043

Adjacent sequences: A063729 A063730 A063731 * A063733 A063734 A063735

Keyword

nonn

Author

Fred Lunnon, Aug 25 2001

Status

approved