%I #21 Jul 13 2024 07:05:39

%S 1,4,7,8,11,14,15,18,21,24,25,28,31,32,35,38,41,42,45,48,49,52,55,56,

%T 59,62,65,66,69,72,73,76,79,82,83,86,89,90,93,96,97,100,103,106,107,

%U 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

%N Positions of 1 in A188374; complement of A188375.

%C See A187950.

%C From _Michel Dekking_, Feb 27 2018: (Start)

%C 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).

%C 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.

%C Proof: recall that

%C 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).

%C 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

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

%C 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.

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

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

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

%C 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

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

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

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

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

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

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

%C (End)

%H Vincenzo Librandi, <a href="/A188376/b188376.txt">Table of n, a(n) for n = 1..2000</a>

%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr.,<a href="http://www.fq.math.ca/Scanned/10-5/carlitz1.pdf"> Pellian Representations</a>, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.

%F a(n) = 2[nr]+n, where r = 1/sqrt(2). - _Michel Dekking_, Feb 27 2018

%t (See A188374.)

%t Table[(2 Floor[n (1/Sqrt[2])] + n), {n, 100}] (* _Vincenzo Librandi_, Mar 01 2018 *)

%o (Magma) [2*Floor(n*(1/Sqrt(2)))+n: n in [1..80]]; // _Vincenzo Librandi_, Mar 01 2018

%Y Cf. A187950, A188374, A188375.

%K nonn

%O 1,2

%A _Clark Kimberling_, Mar 29 2011