|
| |
|
|
A005704
|
|
Number of partitions of 3n into powers of 3.
(Formerly M0639)
|
|
13
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
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] * ... [From Mats Granvik, Gary W. Adamson, Aug 07 2009]
|
|
|
REFERENCES
| G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104-110.
R. K. Guy, personal communication.
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.
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
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
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
|
|
|
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 (benoit7848c(AT)orange.fr), Nov 28 2002
a(n) mod 3 = binomial(2n, n) mod 3. - Benoit Cloitre (benoit7848c(AT)orange.fr), 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]
|
|
|
MATHEMATICA
| Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1, #1}]], #2]]] &, {1}, Range[2, 55]] (* Gyorgy Birkas, 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 *)
|
|
|
CROSSREFS
| Cf. A000041, A000123, A005705, A005706, A018819.
Cf. A006996.
Sequence in context: A117930 A090632 A022786 * A022782 A025692 A137285
Adjacent sequences: A005701 A005702 A005703 * A005705 A005706 A005707
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Formula and more terms from Henry Bottomley (se16(AT)btinternet.com), Apr 30 2001
|
| |
|
|
