A096748 -id:A096748 - 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

A127218 Half-indexed Lucas numbers second version L(n)=A000032=Lucas numbers a(0)=2, a(1)=2, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(2n)=L(n), for n>2: a(2n+1)=L(n)+L(n-3)=2*L(n-1) for n>5: a(n)+a(n+2)=a(n+4) a(2n)=L(n), so a(n)=L(n/2).

Original entry on oeis.org

2, 2, 1, 2, 3, 3, 4, 6, 7, 8, 11, 14, 18, 22, 29, 36, 47, 58, 76, 94, 123, 152, 199, 246, 322, 398, 521, 644, 843, 1042, 1364, 1686, 2207, 2728, 3571, 4414, 5778, 7142, 9349, 11556, 15127, 18698, 24476, 30254, 39603, 48952, 64079

Offset: 0

Miklos Kristof, Mar 28 2007

b(n)=A096748(n-1): for n>5: b(n)+b(n+4)=a(n+2) for n>5: a(n)+a(n+4)=5*b(n+2).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000032, A096748.

b[0]:=2:b[1]:=1:for n from 2 to 80 do b[n]:=b[n-1]+b[n-2] od: a[0]:=2:a[1]:=2:a[2]:=1:a[3]:=2:a[4]:=3:a[5]:=3: for n from 3 to 39 do a[2*n]:=b[n]:a[2*n+1]:=b[n]+b[n-3] od: seq(a[n],n=0..79);

LinearRecurrence[{0,1,0,1},{2,2,1,2,3,3,4,6,7,8},60] (* Harvey P. Dale, Jun 22 2022 *)

Vec((1 + x)*(2 - x^2 + x^3 - x^4 + x^7 - x^8) / (1 - x^2 - x^4) + O(x^45)) \\ Colin Barker, Aug 03 2020

From Colin Barker, Aug 03 2020: (Start)
G.f.: (1 + x)*(2 - x^2 + x^3 - x^4 + x^7 - x^8) / (1 - x^2 - x^4).
a(n) = a(n-2) + a(n-4) for n>10.
(End)

A226649 Fibonacci shuffles: a(2n) = A000071(n) and a(2n+1) = A001611(n+2).

Original entry on oeis.org

0, 2, 0, 3, 1, 4, 2, 6, 4, 9, 7, 14, 12, 22, 20, 35, 33, 56, 54, 90, 88, 145, 143, 234, 232, 378, 376, 611, 609, 988, 986, 1598, 1596, 2585, 2583, 4182, 4180, 6766, 6764, 10947, 10945, 17712, 17710, 28658, 28656, 46369, 46367, 75026, 75024, 121394, 121392, 196419, 196417, 317812, 317810

Offset: 0

V. T. Jayabalaji, Jun 14 2013

a(2*n+1) = a(2*n) + A157725(n); a(2*n) = a(2*n-1) - 2 for n > 0. - Reinhard Zumkeller, Jul 30 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1,1,1).

Cf. A000071, A001611, A096748.

import Data.List (transpose)
a226649 n = a226649_list !! n
a226649_list = concat $ transpose [a000071_list, drop 2 a001611_list]
-- Reinhard Zumkeller, Jul 30 2013

LinearRecurrence[{-1,1,1,1,1},{0,2,0,3,1},60] (* Harvey P. Dale, Sep 12 2018 *)

G.f. -x*(2+x^2+2*x^3+2*x) / ( (1+x)*(x^4+x^2-1) ). - R. J. Mathar, Jul 15 2013
a(n) + a(n+1) = A096748(n+2). - R. J. Mathar, Jul 15 2013
a(2n-1) - 1 = a(2n) + 1 = fib(n+1) = A000045(n+1) for n > 0. - T. D. Noe, Jul 23 2013

A227356 Partial sums of A129361.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 112, 193, 324, 544, 900, 1489, 2442, 4005, 6534, 10660, 17336, 28193, 45760, 74273, 120408, 195200, 316216, 512257, 829458, 1343077, 2174130, 3519412, 5696124, 9219105, 14919408, 24144289

Offset: 1

Kival Ngaokrajang, Jul 08 2013

nonn

Sum of labeled numbers of boxes arranged as Pyramid type-II with base Fibonacci(n).
Let us call a Pyramid "type-I" when each row starts with the same number as the leftmost base number, and "type-II" when each column has the same number as the base.
The Pyramid arrangements are related to other sequences as follows:
   Base Number     Type-I     Type-II
   -----------     ------     -------
   Natural         A002623    A034828
   Odd             A000292    A128624
   Fibonacci       A129696    a(n)
   1               A002620    A002620
   1,0             A008805
See illustration in links.

Kival Ngaokrajang, Illustration for some small n.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,1,-1,0,1).

Cf. A002623, A034828, A002620, A000292, A128624, A129696, A008805.

LinearRecurrence[{2,1,-3,1,-1,0,1},{1,2,5,10,20,36,65},40] (* Harvey P. Dale, Jun 30 2025 *)

For n >=2, a(n) = a(n-1) + A129361(n-1).
G.f. -x*(1+x)*(x^2-x+1) / ( (x-1)*(x^2+x-1)*(x^4+x^2-1) ). - Joerg Arndt, Jul 10 2013
a(n) = 2 + A000045(n+4) - A096748(n+6). - R. J. Mathar, Jul 20 2013

cp's OEIS Frontend

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

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A127218 Half-indexed Lucas numbers second version L(n)=A000032=Lucas numbers a(0)=2, a(1)=2, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(2n)=L(n), for n>2: a(2n+1)=L(n)+L(n-3)=2*L(n-1) for n>5: a(n)+a(n+2)=a(n+4) a(2n)=L(n), so a(n)=L(n/2).

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A226649 Fibonacci shuffles: a(2n) = A000071(n) and a(2n+1) = A001611(n+2).

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A227356 Partial sums of A129361.

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