A277872 Number of ways of writing n as a sum of powers of 4, each power being used at most four times.

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 3, 3, 3, 5, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 5, 3, 3, 3, 4, 1

Offset: 0

Author

Timothy B. Flowers, Nov 03 2016

Keywords

• nonn

Comments

Also known as the hyper 4-ary partition sequence, often denoted h_4(n).

Contains A002487 as a subsequence.

Examples

			a(72) = 4 because 72 = 64+4+4 = 64+4+1+1+1+1 = 16+16+16+16+4+4 = 16+16+16+16+4+1+1+1+1.
		

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.

Crossrefs

Cf. A002487, A054390, A277873.

Programs

Mathematica

n:=250;
r:=3;
(* To get up to n-th term, need r such that 4^r < n < 4^(r+1)  *)
h4 :=  CoefficientList[ Series[ Product[ (1 - q^(5*4^i))/(1 - q^(4^i)) , {i, 0, r}], {q, 0, n} ], q]

Formula

G.f.: Product_{j>=0} (1-x^(5*4^j))/(1-x^(4^j)).

G.f.: Product_{j>=0} (1+x^(4^j)+x^(2*4^j)+x^(3*4^j)+x^(4*4^j)).

a(0)=1 and for n>0, a(4n)=a(n)+a(n-1), a(4n+r)=a(n) for r=1,2,3.

G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4) * A(x^4). - Ilya Gutkovskiy, Jul 09 2019