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%I M0101 N0037 #130 Nov 13 2023 04:46:20
%S 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,
%T 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,
%U 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
%N Number of representations of n as a sum of distinct Fibonacci numbers.
%C Number of partitions into distinct Fibonacci parts (1 counted as single Fibonacci number).
%C Inverse Euler transform of sequence has generating function Sum_{n>1} (x^F(n) - x^(2*F(n)) where F() are the Fibonacci numbers.
%C 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
%C The maxima of the quasi-periods are in A096748. - _Max Barrentine_, Sep 13 2015
%C 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
%D M. Bicknell-Johnson, pp. 53-60 in "Applications of Fibonacci Numbers", volume 8, ed: F. T. Howard, Kluwer (1999); see Theorem 3.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000119/b000119.txt">Table of n, a(n) for n = 0..6765</a>
%H F. Ardila, <a href="https://www.fq.math.ca/Papers1/42-3/quartardila03_2004.pdf">The coefficients of a Fibonacci power series</a>, Fib. Quart. 42 (3) (2004), 202-204.
%H Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/">Home Page</a>
%H Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/2001ExerciceAldo.pdf">An Exercise on Fibonacci Representations</a>, 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.
%H Pierre Bonardo and Anna E. Frid, <a href="https://arxiv.org/abs/1806.09534">Number of valid decompositions of Fibonacci prefixes</a>, arXiv:1806.09534 [math.CO], 2018.
%H Alfred Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 54.
%H L. Carlitz, <a href="https://fq.math.ca/Scanned/6-4/carlitz.pdf">Fibonacci Representations</a>, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220, a(n) = R(N).
%H Sam Chow and Tom Slattery, <a href="https://arxiv.org/abs/2009.08222">On Fibonacci partitions</a>, arXiv:2009.08222 [math.NT], 2020.
%H Sam Chow and Tom Slattery, <a href="https://doi.org/10.1016/j.jnt.2021.02.010">On Fibonacci partitions</a>, Journal of Number Theory, Volume 225, August 2021, Pages 310-326.
%H Sam Chow and Owen Jones, <a href="https://arxiv.org/abs/2308.15415">On the variance of the Fibonacci partition function</a>, arXiv:2308.15415 [math.NT], 2023.
%H Michel Dekking and Ad van Loon, <a href="https://arxiv.org/abs/2304.11387">Counting base phi representations</a>, arXiv:2304.11387 [math.NT], 2023.
%H Tom Kempton, <a href="https://arxiv.org/abs/2311.06006">The Dynamics of the Fibonacci Partition Function</a>, arXiv:2311.06006 [math.NT], 2023.
%H D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, <a href="http://www.fq.math.ca/Scanned/4-4/klarner-a.pdf">part 1</a>, <a href="http://www.fq.math.ca/Scanned/4-4/klarner-b.pdf">part 2</a>, Fib. Quart., 4 (1966), 289-306 and 322.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrep.html#sumoffib">Sumthing about Fibonacci Numbers</a>
%H N. Robbins, <a href="https://www.fq.math.ca/Scanned/34-4/robbins.pdf">Fibonacci partitions</a>, Fib. Quart. 34 (4) (1996), 306-313.
%H Jeffrey Shallit, <a href="https://cs.uwaterloo.ca/~shallit/Papers/ntfl.ps">Number theory and formal languages</a>, 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.)
%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2007.14930">Robbins and Ardila meet Berstel</a>, Arxiv preprint arXiv:2007.14930 [math.CO], 2020.
%H Paul K. Stockmeyer, <a href="http://www.fq.math.ca/Papers1/46_47-2/Stockmeyer.pdf">A Smooth Tight Upper Bound for the Fibonacci Representation Function R(N)</a>, Fibonacci Quarterly, Volume 46/47, Number 2, May 2009.
%H Scott V. Tezlaf, <a href="https://arxiv.org/abs/1806.00331">On ordinal dynamics and the multiplicity of transfinite cardinality</a>, arXiv:1806.00331 [math.NT], 2018. See p. 42.
%F a(A000045(n)) = A065033(n).
%F 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
%F a(n) = 1, if n <= 2; a(n) = a(Fibonacci(i-2)+k)+a(k) if n>2 and 0<=k<Fibonacci(i-3); a(n)= 2*a(k) if n>2 and Fibonacci(i-3)<=k<Fibonacci(i-2); a(n) = a(Fibonacci(i+1)-2-k) otherwise where Fibonacci(i) is the largest Fibonacci number (A000045) <= n and k=n-Fibonacci(i). [Bicknell-Johnson] - _Ron Knott_, Dec 06 2004
%F 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
%F 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
%F a(A000071(n)) = 1. - _Reinhard Zumkeller_, Dec 28 2012
%p 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
%t CoefficientList[ Normal@Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} ], {z, 0, 233} ], z ]
%t nmax = 104; s = Union@Table[Fibonacci[n], {n, nmax}];
%t Table[Length@Select[IntegerPartitions[n, All, s], DeleteDuplicates[#] == # &], {n, 0, nmax}] (* _Robert Price_, Aug 17 2020 *)
%o (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))
%o (PARI) f(x,y,z)=if(x<y, 0^x, f(x-y,y+z,y)+f(x,y+z,y))
%o a(n)=f(n,1,1) \\ _Charles R Greathouse IV_, Dec 14 2015
%o (Haskell)
%o a000119 = p $ drop 2 a000045_list where
%o p _ 0 = 1
%o p (f:fs) m = if m < f then 0 else p fs (m - f) + p fs m
%o -- _Reinhard Zumkeller_, Dec 28 2012, Oct 21 2011
%Y Cf. A007000, A003107, A000121, A080097, A096748. Least inverse is A013583.
%K nonn,nice,look
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 29 2000
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