A107017 -id:A107017 - cp's OEIS frontend

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

Sam Alexander, Oct 19 2003

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

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

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

Partial sums: A130473.

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

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

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 *)

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

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

A139764 Smallest term in Zeckendorf representation of n.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 13, 1, 2, 3, 1, 5, 1, 2, 21, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 34, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 13, 1, 2, 3, 1, 5, 1, 2, 55, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 13, 1, 2, 3, 1, 5, 1, 2, 21, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 89

Offset: 1

Steve Witham (sw(AT)tiac.net), May 15 2008

Also called a "Fibonacci fractal".
Appears to be the same as the "ruler of Fibonaccis" mentioned by Knuth. - N. J. A. Sloane, Aug 03 2012
a(n) is also the number of matches to take away to win in a certain match game (see Rocher et al.).
The frequencies of occurrences of the values in this sequence and A035614 are related by the golden ratio.

			The Zeckendorf representation of 7 = 5 + 2, so a(7) = 2.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 82, solution to Problem 179. - From N. J. A. Sloane, Aug 03 2012

Steve Witham, Table of n, a(n) for n = 1..9999
Alex Bogomolny, Theory of Take-Away Games
Sylvain Rocher, Elodie Privat, Laurent Orban, Alexandre Mothe and Laurent Thouy, La stratégie des allumettes
A. J. Schwenk, Take-Away Games, The Fibonacci Quarterly, v 8, no 3 (1970), 225-234.
Eric Weisstein's World of Mathematics, Wythoff Array
Wikipedia, Zeckendorf's theorem

Cf. A000045, A035614, A107017, A014417, A006519.

Cf. A087172.

a139764 = head . a035517_row  -- Reinhard Zumkeller, Mar 10 2013

A000045 := proc(n) combinat[fibonacci](n) ; end:
A087172 := proc(n)
local a,i ;
a := 0 ;
for i from 0 do
if A000045(i) <= n then
a := A000045(i) ;
else
RETURN(a) ;
fi ;
od:
end:
A139764 := proc(n)
local nResid,prevF ;
nResid := n ;
while true do
prevF := A087172(nResid) ;
if prevF = nResid then
RETURN(prevF) ;
else
nResid := nResid-prevF ;
fi ;
od:
end:
seq(A139764(n),n=1..120) ;
# R. J. Mathar, May 22 2008

f[n_] := (k = 1; ff = {}; While[(fi = Fibonacci[k]) <= n, AppendTo[ff, fi]; k++]; Drop[ff, 1]); a[n_] := First[ If[n == 0, 0, r = n; s = {}; fr = f[n]; While[r > 0, lf = Last[fr]; If[lf <= r, r = r - lf; PrependTo[s, lf]]; fr = Drop[fr, -1]]; s]]; Table[a[n], {n, 1, 89}] (* Jean-François Alcover, Nov 02 2011 *)

a(n)=my(f);forstep(k=log(n*sqrt(5))\log(1.61803)+2, 2, -1, f=fibonacci(k);if(f<=n,n-=f;if(!n,return(f));k--)) \\ Charles R Greathouse IV, Nov 02 2011

a(n) = n if n is a Fibonacci number, else a( n - (largest Fibonacci number < n) ).
a(n) = the value of the (exactly one) digit that turns on between the Fibonacci-base representations of n-1 and n. E.g., from 6 (1001) to 7 (1010), the two's digit turns on.
a(n) = top element of the column of the Wythoff array that contains n.
a(n) = A000045(A035614(n-1) + 2). [Offsets made precise by Peter Munn, Apr 13 2021]
a(n) = A035517(n,0). - Reinhard Zumkeller, Mar 10 2013

More terms from T. D. Noe and R. J. Mathar, May 22 2008

cp's OEIS Frontend

A087172 Greatest Fibonacci number that does not exceed n.

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