

A005704


Number of partitions of 3n into powers of 3.
(Formerly M0639)


22



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
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OFFSET

0,2


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 mary partition function, J. Number Theory 3 (1971), 104110.
G. E. Andrews and J. A. Sellers, Characterizing the number of pary 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 mary partitions, Australasian J. Combin., 30 (2004), 193196.
M. D. Hirschhorn and J. A. Sellers, A different view of mary partitions
D. Krenn, D. Ralaivaosaona, S. Wagner, The Number of MultiBase Representations of an Integer, 2014.
D. Krenn, D. Ralaivaosaona, S. Wagner, MultiBase 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, 285312.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DMCCG), 2001, 215228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DMCCG), 2001, 215228. [Cached copy, with permission]
O. J. Rodseth, Some arithmetical properties of mary partitions, Proc. Camb. Phil. Soc. 68 (1970), 447453.
O. J. Rodseth and J. A. Sellers, On mary partition function congruences: A fresh look at a past problem, J. Number Theory 87 (2001), 270281.
O. J. Rodseth and J. A. Sellers, On a Restricted mNonSquashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.


FORMULA

a(n) = a(n1)+a(floor(n/3)).
Coefficient of x^(3*n) in prod(k>=0, 1/(1x^(3^k))). Also, coefficient of x^n in prod(k>=0, 1/(1x^(3^k)))/(1x).  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)=(1x^3)/(1x)^2*T(x^3). [Joerg Arndt, May 12 2010]


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 *)


PROG

(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A005704(n): return A005704(n1)+A005704(n//3) if n else 1 # Chai Wah Wu, Sep 21 2022


CROSSREFS

Cf. A000041, A000123, A005705, A005706, A018819.
Cf. A006996.
Sequence in context: A117930 A090632 A022786 * A022782 A283968 A025692
Adjacent sequences: A005701 A005702 A005703 * A005705 A005706 A005707


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

Formula and more terms from Henry Bottomley, Apr 30 2001


STATUS

approved



