OFFSET
0,6
COMMENTS
Also known as the hyper 5-ary partition sequence, often denoted h_5(n).
Contains A002487 as a subsequence.
LINKS
Timothy B. Flowers, Table of n, a(n) for n = 0..10000
K. Courtright and J. Sellers, Arithmetic properties for hyper m-ary partition functions, Integers, 4 (2004), A6.
Timothy B. Flowers, Extending a Recent Result on Hyper m-ary Partition Sequences, Journal of Integer Sequences, Vol. 20 (2017), #17.6.7.
T. B. Flowers and S. R. Lockard, Identifying an m-ary partition identity through an m-ary tree, Integers, 16 (2016), A10.
FORMULA
G.f.: Product_{j >= 0} (1-x^(6*5^j))/(1-x^(5^j)).
G.f.: Product_{j >= 0} Sum_{k=0..5} x^(k*5^j).
a(0)=1; for k>0, a(5*k) = a(k)+a(k-1) and a(5*k+r) = a(k) with r=1,2,3,4.
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4 + x^5) * A(x^5). - Ilya Gutkovskiy, Jul 09 2019
EXAMPLE
a(140) = 4 because 140 = 125+5+5+5 = 125+5+5+1+1+1+1+1 = 25+25+25+25+25+5+5+5 = 25+25+25+25+25+5+5+1+1+1+1+1.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
add(b(n-j*5^i, i-1), j=0..min(5, n/5^i))))
end:
a:= n-> b(n, ilog[5](n)):
seq(a(n), n=0..120); # Alois P. Heinz, May 01 2018
MATHEMATICA
n:=250; r:=3; (* To get up to n-th term, need r such that 5^r < n < 5^(r+1) *) h5 := CoefficientList[ Series[ Product[ (1 - q^(6*5^i))/(1 - q^(5^i)) , {i, 0, r}], {q, 0, n} ], q]
CROSSREFS
KEYWORD
nonn
AUTHOR
Timothy B. Flowers, Nov 07 2016
STATUS
approved