A005705 - OEIS

%I M0552 #54 Mar 15 2020 07:50:26

%S 1,2,3,4,6,8,10,12,15,18,21,24,28,32,36,40,46,52,58,64,72,80,88,96,

%T 106,116,126,136,148,160,172,184,199,214,229,244,262,280,298,316,337,

%U 358,379,400,424,448,472,496,524,552,580,608,640,672,704,736,772,808,844

%N Number of partitions of 4*n into powers of 4.

%C Also number of partitions of 4*n+k into powers of 4 where k=1,2,3. - _Michael Somos_, Mar 15 2020

%D R. K. Guy, personal communication.

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

%H Seiichi Manyama, <a href="/A005705/b005705.txt">Table of n, a(n) for n = 0..1000</a>

%H C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, <a href="http://140.109.74.92/hk/wp-content/files/2012/07/mis-n-to-the-logn.pdf">Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics</a>, preprint 2012; SIAM J. Discrete Math., 28(1), 342-371, 2014. - _N. J. A. Sloane_, Dec 23 2012

%H R. K. Guy, <a href="/A000123/a000123_1.pdf">Letters to N. J. A. Sloane and J. W. Moon, 1988</a>

%H M. D. Hirschhorn and J. A. Sellers, <a href="https://ajc.maths.uq.edu.au/pdf/30/ajc_v30_p193.pdf">A different view of m-ary partitions</a>, Australasian J. Combin., 30 (2004), 193-196.

%H M. D. Hirschhorn and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/mike-m-ary.pdf">A different view of m-ary partitions</a>

%H M. Latapy, <a href="https://hal.inria.fr/hal-01182959">Partitions of an integer into powers</a>, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.

%H M. Latapy, <a href="/A005706/a005706.pdf">Partitions of an integer into powers</a>, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]

%H O. J. Rodseth and J. A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Sellers/sellers75.html">On a Restricted m-Non-Squashing Partition Function</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

%F a(n) = a(n-1) + a(floor(n/4)).

%F G.f.: T(x)=(1-x)^(-1)/(Product_{k>=0} 1-x^(4^k)), it satisfies T(x)=(1-x^4)/(1-x)^2*T(x^4). - _Joerg Arndt_, May 12 2010

%t Fold[Append[#1, Total[Take[Flatten[Transpose[Table[#1, {4}]]], #2]]] &, {1},  Range[2, 20]] (* _Birkas Gyorgy_, Apr 18 2011 *)

%Y Column k=4 of A292477.

%Y Cf. A000041, A000123, A005704, A005706.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Formula and more terms from _Henry Bottomley_, Apr 30 2001