A188376 Positions of 1 in A188374; complement of A188375.

1, 4, 7, 8, 11, 14, 15, 18, 21, 24, 25, 28, 31, 32, 35, 38, 41, 42, 45, 48, 49, 52, 55, 56, 59, 62, 65, 66, 69, 72, 73, 76, 79, 82, 83, 86, 89, 90, 93, 96, 97, 100, 103, 106, 107, 110, 113, 114, 117, 120, 123, 124, 127, 130, 131, 134, 137, 140, 141, 144, 147, 148, 151, 154, 155, 158, 161, 164, 165, 168, 171, 172, 175, 178

Offset: 1

Author

Clark Kimberling, Mar 29 2011

Keywords

• nonn

Comments

See A187950.

From Michel Dekking, Feb 27 2018: (Start)

Let d = 3,3,1,3,3,1,3,3,3,1,3,3,1,3,3,3,1,3, ... be the sequence of first differences: d(n):=a(n+1)-a(n).

CLAIM: d equals the Pell word A171588 on the alphabet {3,1}, i.e., d is the unique fixed point of the morphism 3->331, 1->3.

Proof: recall that

A188374 = [nr+2r]-[nr]-[2r] = 1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,... where r=1/sqrt(2).

It was shown in the comments of A294180 that (a(n)) gives the positions of 1 in the 3-symbol Pell word b = A294180 , which is the unique fixed point of the morphism

beta: 1->123, 2->123, 3->1.

The letter 1 occurs in b if and only if it appears as the first letter of a beta(1), beta(2) or beta(3). The differences between the occurrences of 1's are therefore equal to 3, 3, or 1, and moreover, these differences occur exactly as the sequence of beta(1)'s, beta(2)'s and beta(3)'s. After projecting 1->3, 2->3, 3->1 this yields the morphism 3->331, 1->3.

COROLLARY: a(n) = 2[nr] +n.

Proof: We know that the Pell word A171588 = 0010010001001... has a Sturmian representation

A289001(n) = [(n+1)(1-r)]- [n(1-r)] = [nr]-[(n+1)r]-1.

Mapping 0 to 3, and 1 to 1, we find that d = 3313313331331 has a representation d(n) = 2[(n+1)r]-2[nr] +1. This leads to

a(n+1) = 1+d(1)+...+d(n) = n+1+2[(n+1)r].

CLAIM: (a(n)) equals the sequence ad' in the paper "Pellian representations", defined by ad'(n) = [2r[n(1+r)]], for n=1,2,...

Proof: The double floor in the definition of ad' can be reduced to a single floor by Theorem 7.10 of "Pellian representations":

ad'(n) = 2d'(n)-n, for n=1,2,...

Here d' is defined as d'(n) = [n(1+r)]. It follows that

ad'(n) = [n(1+r)]+[nr] = 2[nr]+n = a(n).

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000
• L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.

Crossrefs

Cf. A187950, A188374, A188375.

Programs

Magma

[2*Floor(n*(1/Sqrt(2)))+n: n in [1..80]]; // Vincenzo Librandi, Mar 01 2018

Mathematica

(See A188374.)
Table[(2 Floor[n (1/Sqrt[2])] + n), {n, 100}] (* Vincenzo Librandi, Mar 01 2018 *)

Formula

a(n) = 2[nr]+n, where r = 1/sqrt(2). - Michel Dekking, Feb 27 2018