cp's OEIS Frontend

A287775 Positions of 0 in A287772; complement of A050140 (conjectured and proved).

%I A287775 #19 May 05 2025 16:35:20
%S A287775 2,3,6,7,9,10,13,14,17,18,20,21,24,25,27,28,31,32,35,36,38,39,42,43,
%T A287775 46,47,49,50,53,54,56,57,60,61,64,65,67,68,71,72,74,75,78,79,82,83,85,
%U A287775 86,89,90,93,94,96,97,100,101,103,104,107,108,111,112,114
%N A287775 Positions of 0 in A287772; complement of A050140 (conjectured and proved).
%C A287775 -1 < n*r - a(n) < 1 for n >= 1, where r = (5 + sqrt(5))/4.
%C A287775 From _Michel Dekking_, Dec 28 2017: (Start)
%C A287775 Let (d(n)) be the sequence of first differences: d(n)=a(n+1)-a(n).
%C A287775 CLAIM: d(n) = A108103(n+1) for n=1,2,….
%C A287775 Proof: As a word A287772 = 100110010011001100100110010011... obtained by substituting 0->1, 1->00 in the Fibonacci word F=0100101001001010010100...
%C A287775 This implies that A287772 is a concatenation of 00’s separated by 1’s and 11’s. Moreover, a 0110 occurs iff 1001 occurs in F, and a 010 occurs iff 101 occurs in F. Note also that occurrence of a 00 in A287772 yields a d(n)=1 (and so every other letter in d is a 1), occurrence of a 010 yields a d(n)=2, and occurrence of a 0110 yields a d(n)=3. Since the 1001’s and 101’s occur in 1F according to F itself with 1 prepended (see A001468 and A282162), we must have d(n)=A108103(n+1). (End)
%H A287775 Clark Kimberling, <a href="/A287775/b287775.txt">Table of n, a(n) for n = 1..10000</a>
%t A287775 s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)
%t A287775 w = StringJoin[Map[ToString, s]]
%t A287775 w1 = StringReplace[w, {"0" -> "1", "1" -> "00"}]
%t A287775 st = ToCharacterCode[w1] - 48    (* A287772 *)
%t A287775 Flatten[Position[st, 0]]  (* A287775 *)
%t A287775 Flatten[Position[st, 1]]  (* A050140 conjectured *)
%o A287775 (Python)
%o A287775 from math import isqrt
%o A287775 def A287775(n):
%o A287775     def f(x): return n+(r:=isqrt(x**2//5))+((isqrt(5*(r+1)**2)+r+1&-2)-r-1<=x)
%o A287775     m, k = n, f(n)
%o A287775     while m != k: m, k = k, f(k)
%o A287775     return m # _Chai Wah Wu_, May 05 2025
%Y A287775 Cf. A050140, A287772, A108103.
%K A287775 nonn,easy
%O A287775 1,1
%A A287775 _Clark Kimberling_, Jun 03 2017