|
Search: id:A007895
|
|
|
| A007895 |
|
Number of terms in Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers). |
|
+0
29
|
|
| 0, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 2, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3
(list; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
Equivalently, number of different Fibonacci numbers that sum up to n: n=sum(Fib(i)) i ranging from 1 to a(n). - Carmine Suriano (surianonoi5(AT)libero.it), Apr 20 2009
a(n) differs from sequence A105446 that counts the symbols in the Roman Fibonacci number representation of n. Example: n=12 Number of symbols in the Roman Fibonacci representing 12 is 2 namely 1A; in this sequnce a(12)=3 since 12=1+3+8=Fib(1)+Fib(4)+Fib(6). Of course a(n)=1 whenever n is a Fibonacci. - Carmine Suriano (surianonoi5(AT)libero.it), Apr 20 2009
Also number of 0's (or B's) in the Wythoff representation of n - see the Reble link. See also A135817 for references and links for the Wythoff representation for n>=1. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008; N. J. A. Sloane (njas(AT)research.att.com), Jun 28 2008
Or, a(n) = number of applications of Wythoff's B sequence A001950 needed in the unique Wythoff representation of n>=1. E.g. 16=A(B(A(A(B(1))))) = ABAAB = `10110`, hence a(16)=2. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008
Let M(0)=0, M(1)=1 and for i > 0, M(i+1)=f(concatenation of M(j), j from 0 to i-1) where f is the morphism f(k)=k+1. Then sequence = concatenation of M(j) for j from 0 to infinity. - Claude Lenormand (claude.lenormand(AT)free.fr), Dec 16 2003
|
|
REFERENCES
|
D. E. Daykin, Representation of natural numbers as sums of generalized Fibonacci numbers, J. London Math. Soc. 35 (1960) 143-160.
C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin 29 (1952) 190-195.
F. Weinstein, The Fibonacci Partitions, preprint, 1995.
E. Zeckendorf, Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liege 41, 179-182, 1972.
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..10000
Joerg Arndt, Fxtbook
I. Nemes, Fibonacci representations of multiples of Fibonacci numbers
Don Reble, Zeckendorf vs. Wythoff representations: Comments on A007895
F. V. Weinstein, Notes on Fibonacci partitions
|
|
FORMULA
|
a(n) = A000120(A003714(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 05 2005
a(n) = A107015(n) + A107016(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 09 2005
|
|
EXAMPLE
|
a(46) = a(1+3+8+34) = 4.
|
|
CROSSREFS
|
Cf. A000045, A035514, A035515, A035516, A035517, A105446.
Cf. A135817 (lengths of Wythoff representation), A135818 (number of 1's (or A's) in the Wythoff representation).
Record positions are in A027941.
Adjacent sequences: A007892 A007893 A007894 this_sequence A007896 A007897 A007898
Sequence in context: A085761 A102382 A024890 this_sequence A053260 A140223 A014643
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Felix Weinstein (wain(AT)ana.unibe.ch) and Clark Kimberling (ck6(AT)evansville.edu)
|
|
EXTENSIONS
|
Edited by N. J. A. Sloane (njas(AT)research.att.com) Jun 27 2008 at the suggestion of R. J. Mathar and Don Reble
|
|
|
Search completed in 0.003 seconds
|
