A005706 Number of partitions of 5n into powers of 5.

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, 82, 89, 96, 103, 110, 119, 128, 137, 146, 155, 166, 177, 188, 199, 210, 223, 236, 249, 262, 275, 290, 305, 320, 335, 350, 368, 386, 404, 422, 440, 461, 482, 503, 524, 545

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Euler transform of [2,0,0,0,1,0,0,0,0,...] with 1's at 5^n. - Michael Somos, Mar 16 2004

Partial sums of number of partitions of n into powers of 5. - Michael Somos, Mar 16 2004

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

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• 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, preprint 2012; SIAM J. Discrete Math., 28(1), 342-371, 2014. - _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
• 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 and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

Crossrefs

Column k=5 of A292477.

Cf. A000041, A000123, A005704, A005705.

Programs

Mathematica

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 *)

PARI

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

Formula

a(n) = a(n-1) + a([n/5]).

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

Extensions

Formula and more terms from Henry Bottomley, Apr 30 2001