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A087172
Greatest Fibonacci number that does not exceed n.
11
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
1,2
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
Eric Weisstein's World of Mathematics, Fibonacci Number.
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
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 *)
PROG
(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
(Haskell)
a087172 = head . a035516_row -- Reinhard Zumkeller, Mar 10 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sam Alexander, Oct 19 2003
STATUS
approved