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 A055778 #92 Jul 13 2025 10:59:25
%S A055778 0,1,2,2,3,3,3,2,3,4,4,5,4,4,4,5,4,4,2,3,4,4,5,5,5,4,5,6,6,7,5,5,5,6,
%T A055778 5,5,4,5,6,6,7,5,5,5,6,5,5,2,3,4,4,5,5,5,4,5,6,6,7,6,6,6,7,6,6,4,5,6,
%U A055778 6,7,7,7,6,7,8,8,9,6,6,6,7,6,6,5,6,7,7,8,6,6,6,7,6,6,4,5,6,6,7,7,7,6,7,8,8
%N A055778 Number of 1's in the base-phi representation of n.
%C A055778 Uses greedy algorithm (start with largest possible power of phi, then work downward) - see pseudo-code below.
%C A055778 Conjecture: For all n, A007895(n) <= a(n). There is equality at 1, 7, 18, 19, 47, 48, 54, 123, 124, 130, 141, 142, 322, 323, 329, 340, 341, 369, 370, 376, 843, 844, 850, 861, 862, 890, 891, 897, 966, 967, 973, 984, 985, 2207, 2208, 2214, 2225, 2226, 2254, 2255, 2261, 2330, 2331, 2337, 2348, 2349, 2529, 2530, 2536, 2547, 2548, 2576, 2577, 2583, ... - _Dale Gerdemann_, Apr 01 2012
%C A055778 From _Michel Dekking_, Feb 06 2021: (Start)
%C A055778 Here is a proof that there are infinitely n many such that A007895(n) = a(n).
%C A055778 Let F(n) = A000045(n) be the n-th Fibonacci number, and let L(n) = A000032(n) be the n-th Lucas number.
%C A055778 Then a(L(2k)) = 2, since L(2k) = phi^(2k) + phi^(-2k), where phi is the golden ratio.
%C A055778 On the other hand, A007895(L(2k)) = 2, since L(n) = F(n+1) + F(n-1) for all n > 1. So for k>1 one has A007895(L(2k)) = a(L(2k)) = 2.
%C A055778 (End)
%C A055778 Gerdemann's conjecture above is true: we can run the Bellman-Ford algorithm to determine the lowest-weight path from the initial state to a final state in the weighted directed graph derived from the automaton "saka" in my paper cited below in the Links section, and verify that they all have nonnegative weight. - _Jeffrey Shallit_, May 07 2023
%C A055778 Furthermore, the set of n, in Zeckendorf representation, for which A007895(n) = a(n), is accepted by an 8-state finite automaton. - _Jeffrey Shallit_, May 08 2023
%H A055778 Carmine Suriano, <a href="/A055778/b055778.txt">Table of n, a(n) for n = 0..5000</a>
%H A055778 Michel Dekking, <a href="https://arxiv.org/abs/2003.14125">Points of increase of the sum of digits function of the base phi expansion</a>, arXiv:2003.14125 [math.CO], 2020.
%H A055778 F. Michel Dekking, <a href="https://doi.org/10.1016/j.tcs.2021.01.011">The sum of digits functions of the Zeckendorf and the base phi expansions</a>, Theoretical Computer Science, 2021.
%H A055778 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phigits.html">Using Powers of Phi to represent Integers (Base Phi)</a> (inspiration for this sequence).
%H A055778 Jeffrey Shallit, <a href="https://arxiv.org/abs/2305.02672">Proving Properties of phi-Representations with the Walnut Theorem-Prover</a>, arXiv:2305.02672 [math.NT], 2023.
%H A055778 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PhiNumberSystem.html">Phi Number System</a>
%F A055778 a(n) = delta(x), where x is the fixed point starting with (0,0) of the morphism (j,0)->(j,0)(j,1), (j,1)->(j,2)(j,3), (j,2)->(j+2,0)(j+2,1)(j+2,2), (j,3)->(j+1,3)(j+2,2)(j+1,3) for all natural numbers j, and delta is the decoration morphism (j,0)-> j,j+1, (j,1)-> j+2, (j,2)-> j+2,j+3, (j,3)-> j+3,j+3 for all natural numbers j. - _Michel Dekking_, Feb 06 2021
%F A055778 a(n) <= (A190796(n) + 1)/2. - _Charles R Greathouse IV_, Apr 21 2023
%e A055778 The phi-expansions for n<=15 are:
%e A055778 n phi-rep(n) a(n)
%e A055778 0 0. 0
%e A055778 1 1. 1
%e A055778 2 10.01 2
%e A055778 3 100.01 2
%e A055778 4 101.01 3
%e A055778 5 1000.1001 3
%e A055778 6 1010.0001 3
%e A055778 7 10000.0001 2
%e A055778 8 10001.0001 3
%e A055778 9 10010.0101 4
%e A055778 10 10100.0101 4
%e A055778 11 10101.0101 5
%e A055778 12 100000.101001 4
%e A055778 13 100010.001001 4
%e A055778 14 100100.001001 4
%e A055778 15 100101.001001 5
%e A055778 - _Joerg Arndt_, Jan 30 2012
%t A055778 nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; Total[d[[1]]], {n, 0, nn}] (* _T. D. Noe_, May 20 2011 *)
%o A055778 (Pseudocode)
%o A055778 constant (float): phi=(sqrt(5)+1)/2; function: lphi(x)=log(x)/log(phi); variable (float): rem=n; variable (integer): count=0; loop: while rem>0 {rem=rem-phi^floor[lphi(rem)]; count++;} result: return count; // _Henry Bottomley_, Aug 04 2000
%Y A055778 See also A330037 = (a(n) mod 2).
%Y A055778 Cf. A001622.
%K A055778 base,easy,nonn
%O A055778 0,3
%A A055778 Robert Lozyniak (11(AT)onna.com), Jul 12 2000
%E A055778 More terms from _Henry Bottomley_, Aug 04 2000