A284395 -id:A284395 - cp's OEIS frontend

A284394 {101->2}-transform of the infinite Fibonacci word A003849.

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

Offset: 1

Clark Kimberling, May 02 2017

The sequences p = A032766, q = A284395, 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.

			As a word, A003849 = 01001010010010100..., and replacing each 101 by 2 gives 01002001002002001...

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

Cf. A003849, A032766, A284395, A284396.

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 *)
Flatten[Position[st, 1]]  (* A284395 *)
Flatten[Position[st, 2]]  (* A284396 *)

A284396 Positions of 2 in A284394.

Original entry on oeis.org

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

Clark Kimberling, May 02 2017

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.

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

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.

Cf. A003849, A032766, A284394, A284395.

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 *)

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

cp's OEIS Frontend

A284394 {101->2}-transform of the infinite Fibonacci word A003849.

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