A005704 Number of partitions of 3n into powers of 3.

1, 2, 3, 5, 7, 9, 12, 15, 18, 23, 28, 33, 40, 47, 54, 63, 72, 81, 93, 105, 117, 132, 147, 162, 180, 198, 216, 239, 262, 285, 313, 341, 369, 402, 435, 468, 508, 548, 588, 635, 682, 729, 783, 837, 891, 954, 1017, 1080, 1152, 1224, 1296, 1377, 1458, 1539, 1632

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Infinite convolution product of [1,2,3,3,3,3,3,3,3,3] aerated A000244 - 1 times, i.e., [1,2,3,3,3,3,3,3,3,3] * [1,0,0,2,0,0,3,0,0,3] * [1,0,0,0,0,0,0,0,0,2] * ... [Mats Granvik, Gary W. Adamson, Aug 07 2009]

References

• R. K. Guy, personal communication.
• 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..10000
• G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104-110.
• G. E. Andrews and J. A. Sellers, Characterizing the number of p-ary partitions modulo a prime p, p. 2.
• C. Banderier, H.-K. Hwang, V. Ravelomanana, and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, 2012. - From _N. J. A. Sloane_, Dec 23 2012
• R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
• M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, Australasian J. Combin., 30 (2004), 193-196.
• M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions
• D. Krenn, D. Ralaivaosaona, and S. Wagner, The Number of Multi-Base Representations of an Integer, 2014.
• D. Krenn, D. Ralaivaosaona, and S. Wagner, Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems, arXiv:1503.08594 [math.NT], 2015. Also in Applicable Analysis and Discrete Mathematics (2015) Vol. 9, Issue 2, 285-312.
• M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
• M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
• O. J. Rodseth, Some arithmetical properties of m-ary partitions, Proc. Camb. Phil. Soc. 68 (1970), 447-453.
• O. J. Rodseth and J. A. Sellers, On m-ary partition function congruences: A fresh look at a past problem, J. Number Theory 87 (2001), 270-281.
• O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

Crossrefs

Cf. A000041, A000123, A005705, A005706, A018819.

Cf. A006996.

Programs

Mathematica

Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1, #1}]], #2]]] &, {1}, Range[2, 55]] (* Birkas Gyorgy, Apr 18 2011 *)
a[n_] := a[n] = If[n <= 2, n + 1, a[n - 1] + a[Floor[n/3]]]; Array[a, 101, 0] (* T. D. Noe, Apr 18 2011 *)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A005704(n): return A005704(n-1)+A005704(n//3) if n else 1 # Chai Wah Wu, Sep 21 2022

Formula

a(n) = a(n-1)+a(floor(n/3)).

Coefficient of x^(3*n) in prod(k>=0, 1/(1-x^(3^k))). Also, coefficient of x^n in prod(k>=0, 1/(1-x^(3^k)))/(1-x). - Benoit Cloitre, Nov 28 2002

a(n) mod 3 = binomial(2n, n) mod 3. - Benoit Cloitre, Jan 04 2004

Let T(x) be the g.f., then T(x)=(1-x^3)/(1-x)^2*T(x^3). [Joerg Arndt, May 12 2010]

Extensions

Formula and more terms from Henry Bottomley, Apr 30 2001