A087172 Greatest Fibonacci number that does not exceed n.

1, 2, 3, 3, 5, 5, 5, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55

Offset: 1

Author

Sam Alexander, Oct 19 2003

Keywords

• nonn
• easy

Comments

Also the largest term in Zeckendorf representation of n; starting at Fibonacci positions the sequence is repeated again and again in A107017: A107017(A000045(n)+k) = a(k) with 0 < k < A000045(n-1). - Reinhard Zumkeller, May 09 2005

Fibonacci(n) occurs Fibonacci(n-1) times, for n >= 2. - Benoit Cloitre, Dec 15 2022

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fibonacci Number.

Crossrefs

Cf. A000045, A007895, A035516, A035517, A066628, A107017, A130233, A130234, A256654.

Partial sums: A130473.

Programs

Haskell

a087172 = head . a035516_row -- Reinhard Zumkeller, Mar 10 2013

Maple

with(combinat):
A087172 := proc (n) local j: for j while fibonacci(j) <= n do fibonacci(j) end do: fibonacci(j-1) end proc:
seq(A087172(n), n = 1 .. 40); # Emeric Deutsch, Nov 11 2014
# Alternative
N:= 100: # to get a(n) for n from 1 to N
Fibs:= [seq(combinat:-fibonacci(i), i = 1 .. ceil(log[(1 + sqrt(5))/2](sqrt(5)*N)))]:
A:= Vector(N):
for i from 1 to nops(Fibs)-1 do
  A[Fibs[i] .. min(N,Fibs[i+1]-1)]:= Fibs[i]
od:
convert(A,list); # Robert Israel, Nov 11 2014

Mathematica

With[{rf=Reverse[Fibonacci[Range[10]]]},Flatten[Table[ Select[ rf,n>=#&, 1],{n,80}]]] (* Harvey P. Dale, Dec 08 2012 *)
Flatten[Map[ConstantArray[Fibonacci[#],Fibonacci[#-1]]&,Range[15]]] (* Peter J. C. Moses, May 02 2022 *)

PARI

a(n)=my(k=log(n)\log((1+sqrt(5))/2)); while(fibonacci(k)<=n, k++); fibonacci(k--) \\ Charles R Greathouse IV, Jul 24 2012

Formula

a(n) = Fibonacci(A130233(n)) = Fibonacci(A130234(n+1)-1). - Hieronymus Fischer, May 28 2007

a(n) = A035516(n, 0) = A035517(n, A007895(n)-1). - Reinhard Zumkeller, Mar 10 2013

a(n) = n - A066628(n). - Michel Marcus, Feb 02 2016

Sum_{n>=1} 1/a(n)^2 = Sum_{n>=1} Fibonacci(n)/Fibonacci(n+1)^2 = 1.7947486789... . - Amiram Eldar, Aug 16 2022