

A000119


Number of representations of n as a sum of distinct Fibonacci numbers.
(Formerly M0101 N0037)


63



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
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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 quasiperiods (from Fib(i)1 to Fib(i+1)1, inclusive) is a Fibonacci number + 1. The maximum value of a(n) within each subsequent quasiperiod 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 quasiperiods 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. BicknellJohnson, pp. 5360 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
Jean Berstel, Home Page
J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary.
Pierre Bonardo, 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.
D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289306 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, SpringerVerlag, 1999, pp. 547570. (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.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 42.


FORMULA

a(n) = (1/n)*Sum_{k=1..n} b(k)*a(nk), 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(i2)+k)+a(k) if n>2 and 0<=k<=fib(i3); a(n)= 2*a(k) if n>2 and fib(i3)<=k<=fib(i2); a(n) = a(fib(i+1)2k) otherwise where fib(i) is largest Fibonacci number (A000045) <= n and k=nfib(i). [BicknellJohnson]  Ron Knott, Dec 06 2004
a(n) = f(n,1,1) with f(x,y,z) = if x<y then 0^x else f(xy,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..n1, 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(xy, 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 * A286326 A097368 A271900
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



