OFFSET
1,2
COMMENTS
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.
REFERENCES
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
LINKS
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
FORMULA
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) = A035517(n,0). - Reinhard Zumkeller, Mar 10 2013
EXAMPLE
The Zeckendorf representation of 7 = 5 + 2, so a(7) = 2.
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
(Haskell)
a139764 = head . a035517_row -- Reinhard Zumkeller, Mar 10 2013
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Steve Witham (sw(AT)tiac.net), May 15 2008
EXTENSIONS
More terms from T. D. Noe and R. J. Mathar, May 22 2008
STATUS
approved