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 A095791 #52 Jan 05 2025 19:51:37
%S A095791 1,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,
%T A095791 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U A095791 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9
%N A095791 Number of digits in lazy-Fibonacci-binary representation of n (A104326).
%C A095791 The lazy Fibonacci representation of n >= 0 is obtained by replacing every string of 0's in the binary representation of n by a single 0, thus obtaining a finite zero-one sequence (d(2), d(3), d(4), ..., d(k)), and then forming d(2)*F(2) + d(3)*F(3) + ... + d(k)*F(k), as in the Mathematica program. The lazy Fibonacci representation is often called the maximal Fibonacci representation, in contrast to the Zeckendorf representation, also called the minimal Fibonacci representation. - _Clark Kimberling_, Mar 04 2015
%C A095791 Regarding the References, the lazy Fibonacci representation is sometimes attributed to Erdős and Joo, but it is also found in Brown and Ferns. - _Clark Kimberling_, Mar 04 2015
%H A095791 Amiram Eldar, <a href="/A095791/b095791.txt">Table of n, a(n) for n = 0..10000</a>
%H A095791 J. L. Brown, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-1/brown.pdf">A new characterization of the Fibonacci numbers</a>, Fibonacci Quarterly 3, no. 1 (1965), 1-8.
%H A095791 P. Erdős and I. Joo, <a href="http://dx.doi.org/10.1007/BF02260391">On the Expansion of 1 = Sum{q^(-n_i)}</a>, Period. Math. Hung. 23 (1991), no. 1, 25-28.
%H A095791 H. H. Ferns, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-1/ferns.pdf">On the representation of integers as sums of distinct Fibonacci numbers</a>, Fibonacci Quarterly 3, no. 1 (1965), 21-29.
%H A095791 Clark Kimberling, <a href="https://doi.org/10.4171/EM/468">Intriguing infinite words composed of zeros and ones</a>, Elemente der Mathematik, Vol. 78, No. 1 (2021), pp. 1-8.
%H A095791 Wolfgang Steiner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-1/paper43-1-8.pdf">The joint distribution of greedy and lazy Fibonacci expansions</a>, Fib. Q., 43 (No. 1, 2005), 60-69.
%F A095791 1, 1, then F(3) 2's, then F(4) 3's, then F(5) 4's, ..., then F(k+1) k's, ...
%F A095791 a(0)=a(1)=1 then a(n) = a(floor(n/tau))+1 where tau=(1+sqrt(5))/2. - _Benoit Cloitre_, Dec 17 2006
%F A095791 a(n) is the least k such that f^(k)(n)=0 where f^(k+1)(x)=f(f^(k)(x)) and f(x)=floor(x/phi) where phi=(1+sqrt(5))/2 (see PARI/GP program). - _Benoit Cloitre_, May 24 2007
%F A095791 a(n) = A070939(A104326(n)). - _Amiram Eldar_, Oct 10 2023
%e A095791 The lazy Fibonacci representation of 14 is 8+3+2+1, which in binary notation is 10111, which consists of 5 digits.
%t A095791 t=DeleteCases[IntegerDigits[-1+Range[200],2],{___,0,0,___}];
%t A095791 A181632=Flatten[t]
%t A095791 A095791=Map[Length,t]
%t A095791 A112309=Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]]+1],0]&,t]
%t A095791 A112310=Map[Length,A112309]
%t A095791 (* _Peter J. C. Moses_, Mar 03 2015 *)
%o A095791 (PARI) a(n)=if(n<2,1,a(floor(n*(-1+sqrt(5))/2))+1) \\ _Benoit Cloitre_, Dec 17 2006
%o A095791 (PARI) a(n)=if(n<0,0,c=1;s=n;while(floor(s*2/(1+sqrt(5)))>0,c++;s=floor(s*2/(1+sqrt(5))));c) \\ _Benoit Cloitre_, May 24 2007
%Y A095791 Cf. A000045, A070939, A072649, A095792, A104326, A112309, A112310, A181632.
%K A095791 nonn,base,easy
%O A095791 0,3
%A A095791 _Clark Kimberling_, Jun 05 2004