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A072649 n occurs Fibonacci(n) times (cf. A000045).

%I A072649 #103 Nov 04 2024 11:57:10
%S A072649 1,2,3,3,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,8,
%T A072649 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
%U A072649 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10
%N A072649 n occurs Fibonacci(n) times (cf. A000045).
%C A072649 Number of digits in Zeckendorf-binary representation of n. E.g., the Zeckendorf representation of 12 is 8+3+1, which in binary notation is 10101, which consists of 5 digits. - _Clark Kimberling_, Jun 05 2004
%C A072649 First position where value n occurs is A000045(n+1), i.e., a(A000045(n)) = n-1, for n >= 2 and a(A000045(n)-1) = n-2, for n >= 3.
%C A072649 This is the number of distinct Fibonacci numbers greater than 0 which are less than or equal to n. - _Robert G. Wilson v_, Dec 10 2006
%C A072649 The smallest nondecreasing sequence a(n) such that a(Fibonacci(n-1)) = n. - _Tanya Khovanova_, Jun 20 2007
%H A072649 Reinhard Zumkeller, <a href="/A072649/b072649.txt">Table of n, a(n) for n = 1..10000</a>
%H A072649 A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoff array</a>, arXiv:2405.18128 [math.CO], 2024. See page 4.
%H A072649 Popular Computing (Calabasas, CA), <a href="/A256967/a256967.png">A Coding Exercise (from a suggestion by R. W. Hamming)</a>, Vol. 5 (No. 54, Sep 1977), p. PC55-18.
%F A072649 G.f.: (Sum_{n>1} x^Fibonacci(n))/(1-x). - _Michael Somos_, Apr 25 2003
%F A072649 From _Hieronymus Fischer_, May 02 2007: (Start)
%F A072649 a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) - 1, where phi is A001622.
%F A072649 a(n) = floor(arcsinh(sqrt(5)*n/2)/log(phi)) - 1.
%F A072649 a(n) = A108852(n) - 2. (End)
%F A072649 a(n) = -1 + floor( log_phi( (n+0.2)*sqrt(5) ) ), where log_phi(x) is the logarithm to the base (1+sqrt(5))/2. - _Ralf Stephan_, May 14 2007
%F A072649 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - _Amiram Eldar_, Feb 18 2024
%e A072649 1, 1, then F(2) 2's, then F(3) 3's, then F(4) 4's, ..., then F(k) k's, ...
%p A072649 A072649 := proc(n)
%p A072649     local j;
%p A072649     for j from ilog[(1+sqrt(5))/2](n) while combinat[fibonacci](j+1)<=n do
%p A072649     end do;
%p A072649     j-1
%p A072649 end proc:
%p A072649 seq(A072649(n), n=1..120);  # _Alois P. Heinz_, Mar 18 2013
%t A072649 Table[Table[n, {Fibonacci[n]}], {n, 10}] // Flatten (* _Robert G. Wilson v_, Jan 14 2007 *)
%t A072649 a[n_] := Module[{j}, For[j = Floor@Log[GoldenRatio, n], Fibonacci[j+1] <= n, j++]; j-1];
%t A072649 Table[a[n], {n, 1, 120}] (* _Jean-François Alcover_, Nov 17 2022, after _Alois P. Heinz_ *)
%o A072649 (PARI) a(n) = -1+floor(log(((n+0.2)*sqrt(5)))/log((1+sqrt(5))/2))
%o A072649 (PARI) a(n)=local(m); if(n<1,0,m=0; until(fibonacci(m)>n,m++); m-2)
%o A072649 (Haskell)
%o A072649 a072649 n = a072649_list !! (n-1)
%o A072649 a072649_list = f 1 where
%o A072649    f n = (replicate (fromInteger $ a000045 n) n) ++ f (n+1)
%o A072649 -- _Reinhard Zumkeller_, Jul 04 2011
%o A072649 (Python)
%o A072649 from sympy import fibonacci
%o A072649 def a(n):
%o A072649     if n<1: return 0
%o A072649     m=0
%o A072649     while fibonacci(m)<=n: m+=1
%o A072649     return m-2
%o A072649 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 09 2017
%o A072649 (Python)
%o A072649 def A072649(n):
%o A072649     a, b, c = 0, 1, -2
%o A072649     while a <= n:
%o A072649         a, b = b, a+b
%o A072649         c += 1
%o A072649     return c # _Chai Wah Wu_, Nov 04 2024
%o A072649 (MIT/GNU Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b))))))) ;; (The implementation below is better)
%o A072649 (Scheme) (define (A072649 n) (if (<= n 3) n (let loop ((k 5)) (if (> (A000045 k) n) (- k 2) (loop (+ 1 k)))))) ;; (Use this with the memoized implementation of A000045 given under that entry. No floating point arithmetic is involved). - _Antti Karttunen_, Oct 06 2017
%Y A072649 Cf. A000045, A095791, A095792.
%Y A072649 Cf. A001622 (golden ratio phi), A073010.
%Y A072649 Used to construct A003714. Cf. also A002024, A072643, A072648, A072650.
%Y A072649 Cf. A131234.
%Y A072649 Partial sums: A256966, A256967.
%K A072649 nonn,look
%O A072649 1,2
%A A072649 _Antti Karttunen_, Jun 02 2002
%E A072649 Typo fixed by _Charles R Greathouse IV_, Oct 28 2009