This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014677 #31 Aug 26 2022 10:26:22
%S A014677 1,-1,1,0,-1,1,-1,1,0,-1,1,0,-1,1,-1,1,0,-1,1,-1,1,0,-1,1,0,-1,1,-1,1,
%T A014677 0,-1,1,0,-1,1,-1,1,0,-1,1,-1,1,0,-1,1,0,-1,1,-1,1,0,-1,1,-1,1,0,-1,1,
%U A014677 0,-1,1,-1,1,0,-1,1,0,-1,1,-1,1,0,-1,1,-1,1,0,-1,1,0,-1,1,-1,1,0,-1,1,0,-1,1,-1,1
%N A014677 First differences of A001468.
%C A014677 A001468 is an infinite Fibonacci word with strings of 2's of length A001468(n) delimited by 1's. - _Paul D. Hanna_, Dec 17 2004
%C A014677 c(n):=a(n-1), n >= 1, is -1 if n is a Wythoff B-number from A001950, it is 0 if n=A(B(m)+1) for some m >= 1, with A(k):=A000201(k) (Wythoff A-numbers) and it is +1 if n=A(A(m)+1)=B(m)+1 for some m >= 0, with B(0):=0. - _Wolfdieter Lang_, Oct 13 2006
%C A014677 This sequence is a symbolic sequence as discussed in the paper "Morphisms, Symbolic Sequences, and Their Standard Forms". It can be derived directly from the 2-block morphism induced by the morphism generating A001468. Since A001468 is the Fibonacci word A003849, but on the alphabet {2,1}, with an extra 1 in front, this 2-block morphism has 3-symbol Fibonacci as a fixed point: A270788. The 2-blocks in A001468 are 12, 21, and 22, yielding the differences a(n) = 1, a(n) = -1, and a(n) = 0. In 3-symbol Fibonacci these correspond to the letters 2, 1, and 3. Expressing this coding with pi given by pi(1)=-1, pi(2)=1, pi(3)=0, we obtain the formula below. Wolfdieter Lang's Wythoff description of (a(n)) follows from the corresponding Wythoff description in A270788. - _Michel Dekking_, Dec 30 2019
%H A014677 F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%F A014677 abs(a(n)) = floor(f*ceiling(n/f)) - ceiling(f*floor(n/f)) where f=phi=(1+sqrt(5))/2; for n > 1, abs(a(n)) = A005713(n-1). - _Benoit Cloitre_, Apr 21 2003
%F A014677 G.f. equals the continued fraction: A(x) = [0;1, 1/x, 1/x, 1/x^2, 1/x^3, 1/x^5, 1/x^8, ..., 1/x^Fibonacci(n), ...]. - _Paul D. Hanna_, Dec 17 2004
%F A014677 a(n) = b(n) - b(n-1) with b(n):=A005614(n), n >= 1.
%F A014677 a(n) = pi(A270788(n)), n >= 1, where pi is the letter-to-letter map pi(1)=-1, pi(2)=1, pi(3)=0. - _Michel Dekking_, Dec 30 2019
%o A014677 (Python)
%o A014677 from math import isqrt
%o A014677 def A014677(n): return (n+isqrt(m:=5*(n+2)**2)>>1)-(n+1+isqrt(m-10*n-15)&-2)+(n+isqrt(m-20*n-20)>>1)+1 # _Chai Wah Wu_, Aug 25 2022
%Y A014677 Cf. A001468, A000045. Essentially equal to A270788.
%Y A014677 Cf. A000201, A001950, A005614, A005713.
%K A014677 sign
%O A014677 0,1
%A A014677 _N. J. A. Sloane_, Nov 07 2001