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 A362756 #6 May 02 2023 12:03:00
%S A362756 0,0,1,1,1,2,1,1,1,2,2,2,3,2,2,2,2,1,1,1,2,2,2,3,2,2,2,3,3,3,4,3,3,3,
%T A362756 3,2,2,2,3,3,3,3,2,2,2,2,1,1,1,2,2,2,3,2,2,2,3,3,3,4,3,3,3,3,2,2,2,3,
%U A362756 3,3,4,3,3,3,4,4,4,5,4,4,4,4,3,3,3,4,4
%N A362756 Sum of the bits of the "fractional part" of the base-phi representation of n.
%C A362756 The phi-representation of n is the (essentially) unique way to write n = Sum_{j=L..R} b(j)*phi^j, where b(j) is in {0,1} and -oo < L <= 0 <= R, where phi = (1+sqrt(5))/2, subject to the condition that b(j)b(j+1) != 1. The "fractional" part is the string of bits b(L)...b(-1).
%C A362756 The first difference of a(n) is Fibonacci-automatic and takes values in {-1,0,1} only.
%H A362756 George Bergman, <a href="https://math.berkeley.edu/~gbergman/papers/base_tau.pdf">A number system with an irrational base</a>, Math. Mag. 31 (1957), 98-110.
%F A362756 There is a linear representation of rank 21 for a(n).
%e A362756 For n = 20 the phi-representation is 1000010.010001, so a(20) = 2.
%Y A362756 Cf. A055778, A362716.
%K A362756 nonn
%O A362756 0,6
%A A362756 _Jeffrey Shallit_, May 02 2023