login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000119 Number of representations of n as a sum of distinct Fibonacci numbers.
(Formerly M0101 N0037)
52
1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 3, 1, 3, 3, 2, 4, 2, 3, 3, 1, 4, 3, 3, 5, 2, 4, 4, 2, 5, 3, 3, 4, 1, 4, 4, 3, 6, 3, 5, 5, 2, 6, 4, 4, 6, 2, 5, 5, 3, 6, 3, 4, 4, 1, 5, 4, 4, 7, 3, 6, 6, 3, 8, 5, 5, 7, 2, 6, 6, 4, 8, 4, 6, 6, 2, 7, 5, 5, 8, 3, 6, 6, 3, 7, 4, 4, 5, 1, 5, 5, 4, 8, 4, 7, 7, 3, 9, 6, 6, 9, 3, 8, 8, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of partitions into distinct Fibonacci parts (1 counted as single Fibonacci number).

Inverse Euler transform of sequence has generating function sum_{n>1} x^F(n)-x^{2F(n)} where F() are the Fibonacci numbers.

A065033(n) = a(A000045(n)).

a(n) = 1 if and only if n+1 is a Fibonacci number. The lengths of such quasi-periods (from Fib(i)-1 to Fib(i+1)-1, inclusive) is a Fibonacci number + 1. The maximum value of a(n) within each subsequent quasi-period increases by a Fibonacci number. For example, from n = 143 to n = 232, the maximum is 13. From 232 to 376, the maximum is 16, an increase of 3. From 376 to 609, 21, an increase of 5. From 609 to 986, 26, increasing by 5 again. Each two subsequent maxima seem to increase by the same increment, the next Fibonacci number. - Kerry Mitchell, Nov 14 2009

a(A000071(n)) = 1. - Reinhard Zumkeller, Dec 28 2012

The maxima of the quasi-periods are in A096748. - Max Barrentine, Sep 13 2015

Stockmeyer proves that a(n) <= sqrt(n+1) with equality iff n = fibonacci(m)^2 - 1 for some m>=2 (cf. A080097). - Michel Marcus, Mar 02 2016

REFERENCES

M. Bicknell-Johnson, pp. 53-60 in 'Applications of Fibonacci Numbers', volume 8, ed: F T Howard, Kluwer (1999); see Theorem 3.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 54.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..6765

Jean Berstel, Home Page

J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.

D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322.

Ron Knott Sumthing about Fibonacci Numbers

J. Shallit, Number theory and formal languages, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570. (Eq. 9.2.)

Paul K. Stockmeyer, A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N), Fibonacci Quarterly, Volume 46/47, Number 2, May 2009.

FORMULA

a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), b(k) = Sum_{f} (-1)^(k/f+1)*f, where the last sum is taken over all Fibonacci numbers f dividing k. - Vladeta Jovovic, Aug 28 2002

a(n) = 1, if n <= 2; a(n) = a(fib(i-2)+k)+a(k) if n>2 and 0<=k<=fib(i-3); a(n)= 2*a(k) if n>2 and fib(i-3)<=k<=fib(i-2); a(n) = a(fib(i+1)-2-k) otherwise where fib(i) is largest Fibonacci number (A000045) <= n and k=n-fib(i). [Bicknell-Johnson] - Ron Knott, Dec 06 2004

a(n) = f(n,1,1) with f(x,y,z) = if x<y then 0^x else f(x-y,y+z,y)+f(x,y+z,y). - Reinhard Zumkeller, Nov 11 2009

G.f.: prod(n>=1, 1 + q^F(n+1) ) = 1 + sum(n>=1, q^F(n+1) * prod(k=1..n-1, 1+ q^F(k+1) ) ). - Joerg Arndt, Oct 20 2012

MAPLE

with(combinat): p := product((1+x^fibonacci(i)), i=2..25): s := series(p, x, 1000): for k from 0 to 250 do printf(`%d, `, coeff(s, x, k)) od: # James A. Sellers, May 29 2000

MATHEMATICA

CoefficientList[ Normal@Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} ], {z, 0, 233} ], z ]

PROG

(PARI) a(n)=local(A, m, f); if(n<0, 0, A=1+x*O(x^n); m=2; while((f=fibonacci(m))<=n, A*=1+x^f; m++); polcoeff(A, n))

(PARI) f(x, y, z)=if(x<y, 0^x, f(x-y, y+z, y)+f(x, y+z, y))

a(n)=f(n, 1, 1) \\ Charles R Greathouse IV, Dec 14 2015

(Haskell)

a000119 = p $ drop 2 a000045_list where

   p _      0 = 1

   p (f:fs) m = if m < f then 0 else p fs (m - f) + p fs m

-- Reinhard Zumkeller, Dec 28 2012, Oct 21 2011

CROSSREFS

Cf. A007000, A003107, A000121, A080097, A096748. Least inverse is A013583.

Sequence in context: A152545 A256660 A109967 * A097368 A271900 A194083

Adjacent sequences:  A000116 A000117 A000118 * A000120 A000121 A000122

KEYWORD

nonn,nice,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, May 29 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 23 22:30 EDT 2017. Contains 283985 sequences.