A284396 Positions of 2 in A284394.

5, 11, 14, 20, 26, 29, 35, 38, 44, 50, 53, 59, 65, 68, 74, 77, 83, 89, 92, 98, 101, 107, 113, 116, 122, 128, 131, 137, 140, 146, 152, 155, 161, 167, 170, 176, 179, 185, 191, 194, 200, 203, 209, 215, 218, 224, 230, 233, 239, 242, 248, 254, 257, 263, 266, 272

Offset

1,1

Comments

The sequences p = A032766, q = A285395, r = A284396 of positions of 0,1,2, respectively, partition the positive integers. Let t,u,v be the slopes of p, q, r, respectively. Then t = 3/2, u = (9+3*sqrt(5))/2, v = (3+3*sqrt(5))/2, and 1/t + 1/u + 1/v = 1.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424v3 [math.NT], 2018-2019.

Formula

a(n) = 3*floor(n*phi) + 2. This follows from Theorem 29 in Allouche and Dekking, since the overlap word 10101 that contains 101 does not occur in the Fibonacci word. Note that v = 3*phi. - Michel Dekking, Oct 17 2018

Example

As a word, A284394 = 01002001002002001..., in which the positions of 2 are 5,11,14,...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13]  (* A003849 *)
w = StringJoin[Map[ToString, s]]; w1 = StringReplace[w, {"101" -> "2"}]
st = ToCharacterCode[w1] - 48 (* A284394 *)
Flatten[Position[st, 0]]  (* A032766, conjectured *)
Flatten[Position[st, 1]]  (* A284395 *)
Flatten[Position[st, 2]]  (* A284396 *)

Crossrefs

Cf. A003849, A032766, A284394, A284395.

Sequence in context: A174177 A256785 A084864 * A313999 A267298 A102801

Adjacent sequences: A284393 A284394 A284395 * A284397 A284398 A284399

Keyword

nonn,easy

Author

Clark Kimberling, May 02 2017

Status

approved