A003107 -id:A003107 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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.

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

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

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

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 *)

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))

f(x,y,z)=if(xCharles R Greathouse IV, Dec 14 2015

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

More terms from James Sellers, May 29 2000

A319394 Number T(n,k) of partitions of n into exactly k positive Fibonacci numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 0, 1, 3, 2, 2, 1, 1, 0, 1, 1, 2, 4, 2, 2, 1, 1, 0, 0, 1, 3, 3, 4, 2, 2, 1, 1, 0, 0, 2, 2, 4, 4, 4, 2, 2, 1, 1, 0, 0, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1, 0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1

Offset: 0

Alois P. Heinz, Sep 18 2018

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(n,k) = 0 for k > n.

			T(14,3) = 2: 851, 833.
T(14,4) = 5: 8321, 8222, 5531, 5522, 5333.
T(14,5) = 6: 83111, 82211, 55211, 53321, 53222, 33332.
T(14,6) = 8: 821111, 551111, 533111, 532211, 522221, 333311, 333221, 332222.
T(14,7) = 7: 8111111, 5321111, 5222111, 3332111, 3322211, 3222221, 2222222.
T(14,8) = 6: 53111111, 52211111, 33311111, 33221111, 32222111, 22222211.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 0, 2, 1, 1;
  0, 1, 1, 2, 1, 1;
  0, 0, 2, 2, 2, 1, 1;
  0, 0, 1, 3, 2, 2, 1, 1;
  0, 1, 1, 2, 4, 2, 2, 1, 1;
  0, 0, 1, 3, 3, 4, 2, 2, 1, 1;
  0, 0, 2, 2, 4, 4, 4, 2, 2, 1, 1;
  0, 0, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1;
  0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1;
  0, 1, 1, 2, 4, 7, 6, 6, 4, 4, 2, 2, 1, 1;
  0, 0, 1, 2, 5, 6, 8, 7, 6, 4, 4, 2, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010056 (for n>0), A319395, A319396, A319397, A319398, A319399, A319400, A319401, A319402, A319403.
Row sums give A003107.
T(2n,n) gives A136343.
Cf. A000045, A281689, A319503.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

T[n_, k_] := SeriesCoefficient[1/Product[(1 - y x^Fibonacci[j]) + O[x]^(n+1) // Normal, {j, 2, n+1}], {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 0, 40}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 28 2020 *)
h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
T[n_, k_] :=  b[n, h[n], k] - b[n, h[n], k - 1];
Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A000045(j)).
Sum_{k=1..n} k * T(n,k) = A281689(n).
T(A000045(n),n) = A319503(n).

A098641 Number of partitions of the n-th Fibonacci number into Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 14, 41, 157, 803, 5564, 53384, 718844, 13783708, 380676448, 15298907733, 902438020514, 78720750045598, 10220860796171917, 1986422867300209784, 580763241873718042562, 256553744608217295298827, 171912553856721407543178940, 175350753369071026461010505478

Offset: 0

Marcel Dubois de Cadouin (dubois.ml(AT)club-internet.fr), Oct 27 2004

nonn

a(n) = A003107(A000045(n)).

			n=6: A000045(6)=8, a(6) = #{8, 5+3, 5+2+1, 5+1+1+1, 3+3+2, 3+3+1+1, 3+2+2+1, 3+2+1+1+1, 3+1+1+1+1+1, 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1} = 14; the other partitions of 8 into parts with at least one non-Fibonacci number: 7+1, 6+2, 6+1+1, 4+4, 4+3+1, 4+2+2, 4+2+1+1 and 4+1+1+1+1.

Alois P. Heinz, Table of n, a(n) for n = 0..27

Cf. A000041, A000045, A065033, A072214, A098642, A098643, A098644, A319503.

cl = CoefficientList[ Series[1/Product[(1 - x^Fibonacci[i]), {i, 2, 21}], {x, 0, 10950}], x]; cl[[ Table[ Fibonacci[i] + 1, {i, 21}] ]] (* Robert G. Wilson v, Apr 25 2005 *)

a(n) = A098642(n) + A098643(n) + A098644(n).

Corrected and extended by Reinhard Zumkeller, Apr 24 2005
a(15)-a(21) from Robert G. Wilson v, Apr 25 2005
Entry revised by N. J. A. Sloane, Mar 29 2006
a(0), a(22)-a(23) from Alois P. Heinz, Sep 20 2018

A007000 Number of partitions of n into Fibonacci parts (with 2 types of 1).

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 35, 49, 66, 88, 115, 148, 189, 238, 297, 368, 451, 550, 665, 799, 956, 1136, 1344, 1583, 1855, 2167, 2520, 2920, 3373, 3882, 4455, 5097, 5814, 6617, 7509, 8502, 9604, 10823, 12173, 13662, 15302, 17110, 19093, 21271, 23657, 26266

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

			a(2)=4 because we have [2],[1',1'],[1',1],[1,1] (the two types of 1 are denoted 1 and 1').

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
James Propp and N. J. A. Sloane, Email, March 1994.

Cf. A003107.

Cf. A000045.

import Data.MemoCombinators (memo2, integral)
a007000 n = a007000_list !! n
a007000_list = map (p' 1) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m | m < fib   = 0
         | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k
-- Reinhard Zumkeller, Dec 09 2015

with(combinat): gf := 1/product((1-q^fibonacci(k)), k=1..20): s := series(gf, q, 200): for i from 0 to 199 do printf(`%d,`,coeff(s, q, i)) od: # James Sellers, Feb 08 2002

CoefficientList[ Series[ 1/Product[1 - x^Fibonacci[i], {i, 1, 15}], {x, 0, 50}], x]
nmax = 46; f = Table[Fibonacci[n], {n, nmax}];
Table[Length[IntegerPartitions[n, All, f]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

a(n) = 1/n*Sum_{k=1..n} (A005092(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Aug 22 2002
G.f.: 1/Product_{j>=1} (1-x^fibonacci(j)). - Emeric Deutsch, Mar 05 2006
G.f.: Sum_{i>=0} x^Fibonacci(i) / Product_{j=1..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017

More terms from James Sellers, Feb 08 2002

A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A384912 at n = 64.

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are Fibonacci numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A003107, A115063, A115975.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384914, A384915, A384916.

fib[n_] := Boole[Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]];
s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * fib[d], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*isfib(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A003107(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.05893526314055968638..., where f(x) = (1-x) / Product_{k>=2} (1-x^A000045(k)).

A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 8, 8, 7, 8, 9, 7, 10, 8, 8, 9, 7, 8, 8, 4, 8, 5, 8, 9, 6, 10, 8, 6, 10, 6, 9, 8, 5, 9, 6, 6, 8, 4, 8, 4, 1, 8, 4, 7, 9, 5, 10, 7, 6, 10, 6, 8, 6, 3, 10, 5, 7, 9, 5, 8, 5, 2, 9, 4, 7, 6, 2, 8, 4, 3, 8, 1, 4, 1

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000045, A003107, A319394, A319397, A359511, A359512, A359516.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..4))(b(n, h(n))):
seq(a(n), n=0..87);  # Alois P. Heinz, Jan 03 2023

h[n_] := h[n] = If[n < 1, 0, With[{t = 5 n^2}, If[IntegerQ @ Sqrt[t + 4] || IntegerQ @ Sqrt[t - 4], n, h[n - 1]]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[x*b[n - i, h[Min[n - i, i]]]]];
a[n_] := Sum[Coefficient[#, x, i], {i, 0, 4}]&[b[n, h[n]]];
Table[a[n], {n, 0, 87}] (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..4} A319394(n,k). - Alois P. Heinz, Jan 03 2023

A089197 Nonadjacent Fibonacci currency: number of ways to make change for n units in a currency system with coins of value 1, 2, 5, 13, 34, 89, ..., Fibonacci(2k-1).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 20, 22, 25, 28, 31, 35, 38, 42, 46, 50, 55, 60, 65, 71, 76, 83, 89, 96, 103, 111, 119, 128, 136, 146, 156, 167, 178, 189, 201, 214, 227, 241, 255, 270, 286, 302, 319, 337, 355, 375, 394, 415, 436, 458, 481, 505, 529, 555

Offset: 0

Wouter Meeussen, Dec 08 2003

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000045, A003107.

Mathematica
```
<
				
```

G.f.: 1/((1-x^1)*(1-x^2)*(1-x^5)*(1-x^13)*(1-x^34)*(1-x^89)*...).

Incorrect comment deleted by Peter Munn, Nov 14 2022

A102848 Number of partitions of n into Fibonacci number of integer parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 29, 37, 47, 59, 74, 92, 114, 141, 173, 213, 261, 318, 387, 470, 569, 687, 827, 994, 1192, 1426, 1702, 2028, 2412, 2863, 3392, 4012, 4738, 5585, 6574, 7726, 9067, 10624, 12433, 14528, 16957, 19763, 23007, 26749, 31067, 36034

Offset: 0

Lior Manor, Feb 28 2005

A003107 & this sequence are different sequences. A003107 gives the number of partitions in which each part of n is a Fibonacci number, this sequence gives the number of partitions in which the number of parts is a Fibonacci number. Both sequences share the same values for the first 9 values. For example A003107(4) = 4 because of the following 4 partitions of 5: (3,1), (2,2), (2,1,1), (1,1,1,1) whereas a(4) is also 4 but because of different set of partitions: (4), (3,1), (2,2), (2,1,1).

			a(5) = 6 since out of 7 possible partitions of 5 into integer parts, only 6 include a Fibonacci number of parts: (5), (4,1), (3,2), (3,1,1), (2,2,1), (1,1,1,1,1). The 7th integer partitions of 5 (2,1,1,1) is not counted since it includes 4 integer parts and 4 is not a Fibonacci number.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000040, A000045, A003107.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1,
      `if`((h-> issqr(h+4) or issqr(h-4))(5*(t+n)^2), 1, 0),
         b(n, i-1, t) + b(n-i, min(i, n-i), t+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Jul 29 2017

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, If[IntegerQ @ Sqrt[# + 4] || IntegerQ @ Sqrt[# - 4]&[5*(t + n)^2], 1, 0], b[n, i - 1, t] + b[n - i, Min[i, n - i], t + 1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

G.f.: 1 + Sum_{n>=2} x^Fibonacci(n)/Product_{i=1..Fibonacci(n)} (1-x^i). - Vladeta Jovovic, Mar 02 2005

More terms from Vladeta Jovovic, Mar 02 2005

a(0)=1 prepended by Alois P. Heinz, Jul 29 2017

A238999 Number of partitions of n using Fibonacci numbers > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 2, 4, 3, 5, 5, 6, 8, 8, 10, 12, 12, 16, 16, 19, 23, 23, 28, 31, 33, 40, 41, 47, 53, 56, 64, 69, 75, 86, 89, 101, 109, 117, 131, 139, 151, 168, 175, 195, 208, 223, 245, 259, 280, 304, 320, 350, 370, 397, 430, 452, 488, 521, 550, 596, 626

Offset: 0

Clark Kimberling, Mar 08 2014

			a(12) counts these partitions:  822, 552, 5322, 3333, 33222, 222222.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003107, A000045, A239000.

p[n_] := IntegerPartitions[n, All, Fibonacci@Range[3, 60]]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80] (*counts partitions, A238999*)

N=66; q='q+O('q^N); Vec( 1/prod(n=1,11,1-q^fibonacci(n+2)) ) \\ Joerg Arndt, Mar 11 2014

G.f.: 1/Product_{i>=3} (1 - x^Fibonacci(i)).

cp's OEIS Frontend

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

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A319394 Number T(n,k) of partitions of n into exactly k positive Fibonacci numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A098641 Number of partitions of the n-th Fibonacci number into Fibonacci numbers.

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A007000 Number of partitions of n into Fibonacci parts (with 2 types of 1).

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A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

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PARI

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A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).

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A089197 Nonadjacent Fibonacci currency: number of ways to make change for n units in a currency system with coins of value 1, 2, 5, 13, 34, 89, ..., Fibonacci(2k-1).

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