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

0,5

Comments

Also known as the hyper 4-ary partition sequence, often denoted h_4(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^(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

Example

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.

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]

Crossrefs

Cf. A002487, A054390, A277873.

Sequence in context: A115362 A365632 A340853 * A161100 A342007 A341997

Adjacent sequences: A277869 A277870 A277871 * A277873 A277874 A277875

Keyword

nonn,changed

Author

Timothy B. Flowers, Nov 03 2016

Status

approved