A284625 Positions of 2 in A284749.

3, 6, 7, 10, 13, 14, 17, 18, 21, 24, 25, 28, 31, 32, 35, 36, 39, 42, 43, 46, 47, 50, 53, 54, 57, 60, 61, 64, 65, 68, 71, 72, 75, 78, 79, 82, 83, 86, 89, 90, 93, 94, 97, 100, 101, 104, 107, 108, 111, 112, 115, 118, 119, 122, 123, 126, 129, 130, 133, 136, 137

Offset: 1

Author

Clark Kimberling, May 02 2017

Keywords

• nonn
• easy

Comments

This sequence and A214971 and A284624 partition the positive integers into sequences with slopes t = (5+sqrt(5))/2, u = (5+sqrt(5))/2, v = sqrt(5), where 1/t + 1/u + 1/v = 1. The positions of 0 in A284749 are given by A214971, and of 1, by A284624.

Examples

			As a word, A284749 = 012012201201220122..., in which 2 is in positions 3,6,7,10,...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424v3 [math.NT], 2018-2019.

Crossrefs

Cf. A214971, A284625, A284749.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13]  (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"001" -> "2"}]
st = ToCharacterCode[w1] - 48 (* A284749 *)
Flatten[Position[st, 0]]  (* A214971 *)
Flatten[Position[st, 1]]  (* A284624 *)
Flatten[Position[st, 2]]  (* A284625 *)

Python

from math import isqrt
def A284625(n): return (n+isqrt(5*n**2)&-2)-n+2 # Chai Wah Wu, May 22 2025

Formula

a(n) = 2*floor(n*phi) - n + 2 (Example 30 in Allouche and Dekking). - Michel Dekking, Oct 08 2018

a(n) = A050140(n)+2 - Michel Dekking, Oct 08 2018