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A005706 Number of partitions of 5n into powers of 5.
(Formerly M0519)

%I M0519

%S 1,2,3,4,5,7,9,11,13,15,18,21,24,27,30,34,38,42,46,50,55,60,65,70,75,

%T 82,89,96,103,110,119,128,137,146,155,166,177,188,199,210,223,236,249,

%U 262,275,290,305,320,335,350,368,386,404,422,440,461,482,503,524,545

%N Number of partitions of 5n into powers of 5.

%C Euler transform of [2,0,0,0,1,0,0,0,0,...] with 1's at 5^n. - _Michael Somos_, Mar 16 2004

%C Partial sums of number of partitions of n into powers of 5. - _Michael Somos_, Mar 16 2004

%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 Alois P. Heinz, <a href="/A005706/b005706.txt">Table of n, a(n) for n = 0..10000</a>

%H C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, <a href="">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([n/5]).

%F a(n) = [x^(5*n)] Product_{k>=0} 1/(1 - x^(5^k)). - _Ilya Gutkovskiy_, Jun 05 2017

%t a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/5]]; Table[a@ n, {n, 0, 60}] (* _Michael De Vlieger_, Mar 25 2016 *)

%o (PARI) a(n)=if(n<1,n==0,a(n-1)+a(n\5))

%Y Column k=5 of A292477.

%Y Cf. A000041, A000123, A005704, A005705.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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

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Last modified May 25 19:13 EDT 2019. Contains 323576 sequences. (Running on oeis4.)