A000119 Number of representations of n as a sum of distinct Fibonacci numbers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• look

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^(2*F(n))) where F() are the Fibonacci numbers.

a(n) = 1 if and only if n+1 is a Fibonacci number. The length of such a quasi-period (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

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.
• 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
• Federico Ardila, The coefficients of a Fibonacci power series, Fib. Quart. 42 (3) (2004), 202-204.
• Erik Bates, Blan Morrison, Mason Rogers, Arianna Serafini, and Anav Sood, A new combinatorial interpretation of partial sums of m-step Fibonacci numbers, arXiv:2503.11055 [math.CO], 2025. See p. 2.
• Jean Berstel, Home Page
• Jean 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.
• Pierre Bonardo and Anna E. Frid, Number of valid decompositions of Fibonacci prefixes, arXiv:1806.09534 [math.CO], 2018.
• Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 54.
• Leonard Carlitz, Fibonacci Representations, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220, a(n) = R(N).
• Sam Chow and Tom Slattery, On Fibonacci partitions, arXiv:2009.08222 [math.NT], 2020.
• Sam Chow and Tom Slattery, On Fibonacci partitions, Journal of Number Theory, Volume 225, August 2021, Pages 310-326.
• Sam Chow and Owen Jones, On the variance of the Fibonacci partition function, arXiv:2308.15415 [math.NT], 2023.
• Michel Dekking and Ad van Loon, Counting base phi representations, arXiv:2304.11387 [math.NT], 2023.
• Tom Kempton, The Dynamics of the Fibonacci Partition Function, arXiv:2311.06006 [math.NT], 2023.
• David 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
• Neville Robbins, Fibonacci partitions, Fib. Quart. 34 (4) (1996), 306-313.
• Jeffrey 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.)
• Jeffrey Shallit, Robbins and Ardila meet Berstel, Arxiv preprint arXiv:2007.14930 [math.CO], 2020.
• Paul K. Stockmeyer, A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N), Fibonacci Quarterly, Volume 46/47, Number 2, May 2009.
• Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 42.

Crossrefs

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

Programs

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

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 Sellers, May 29 2000

Mathematica

CoefficientList[ Normal@Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} ], {z, 0, 233} ], z ]
nmax = 104; s = Union@Table[Fibonacci[n], {n, nmax}];
Table[Length@Select[IntegerPartitions[n, All, s], DeleteDuplicates[#] == # &], {n, 0, nmax}] (* Robert Price, Aug 17 2020 *)

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(xCharles R Greathouse IV, Dec 14 2015

Formula

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

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(Fibonacci(i-2)+k)+a(k) if n>2 and 0<=k2 and Fibonacci(i-3)<=kA000045) <= n and k=n-Fibonacci(i). [Bicknell-Johnson] - Ron Knott, Dec 06 2004

a(n) = f(n,1,1) with f(x,y,z) = if xReinhard Zumkeller, Nov 11 2009

G.f.: Product_{n>=1} 1 + q^F(n+1) = 1 + Sum_{n>=1} ( q^F(n+1) * Product_{k=1..n-1} 1 + q^F(k+1) ). - Joerg Arndt, Oct 20 2012

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

Extensions

More terms from James Sellers, May 29 2000