0,4

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. [From Kerry Mitchell (lkmitch(AT)gmail.com), Nov 14 2009]

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 60-th anniversary.

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.

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

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).

Paul K. Stockmeyer, "A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N)", Fibonacci Quarterly, Volume 46/47, Number 2, May 2009. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]

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

Jean Berstel, Home Page

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.)

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 (vladeta(AT)eunet.rs), Aug 28 2002

a(n)= 1, if n=0, 1, 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 (ron(AT)ronknott.com), 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). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 11 2009]

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:

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

(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))

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

Sequence in context: A160696 A152545 A109967 this_sequence A097368 A109699 A029283

Adjacent sequences: A000116 A000117 A000118 this_sequence A000120 A000121 A000122

nonn,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000