A064630 Number of parts if 4^n is partitioned into parts of size 3^n as far as possible into parts of size 2^n as far as possible and into parts of size 1^n.

2, 5, 5, 16, 25, 15, 66, 121, 146, 771, 1220, 3641, 8093, 15843, 28359, 50236, 33366, 36709, 145250, 137776, 548024, 2186496, 1066102, 4251976, 16984368, 28678103, 13620614, 205950171, 100716646, 381399635, 1397934923, 3826001641

Offset: 1

Author

Labos Elemer, Oct 01 2001

Keywords

• nonn

Comments

Corresponds to the only solution of the Diophantine equation 4^n = x*3^n + y*2^n + z*1^n with constraints 0 <= y < 3^n/2^n, 0 <= z < 2^n.

Binary order (cf. A029837) of a(n) is close to n.

Examples

			4^6 = 4096 = 729 + 729 + 729 + 729 + 729 + 64 + 64 + 64 + 64 + 64 + 64 + 64 + 1 + 1 + 1 = 5*3^6 + 7*2^6 + 3*1^6, so a(6) = 5 + 7 + 3 = 15.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A064628, A064629, A060692, A029837.

Programs

PARI

{for(n=1,32,a=divrem(4^n,3^n); b=divrem(a[2],2^n); print1(a[1]+b[1]+b[2],","))}

PARI

{ f=t=w=1; for (n=1, 250, f*=4; t*=3; w*=2; a=divrem(f, t); b=divrem(a[2], w); write("b064630.txt", n, " ", a[1]+b[1]+b[2]) ) } \\ Harry J. Smith, Sep 20 2009

Formula

a(n) = A064628(n) + floor(A064629(n)/2^n) + (A064629(n) mod 2^n) = floor(4^n/3^n) + floor((4^n mod 3^n)/2^n) + ((4^n mod 3^n) mod 2^n)

Extensions

Edited by Klaus Brockhaus, May 24 2003