A160383 Number of 3's in base-4 representation of n.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Author

Frank Ruskey, Jun 05 2009

Keywords

• nonn
• base
• easy

Links

• F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.

Crossrefs

Cf. A007090 (base 4), A160380 (0's), A160381 (1's), A160382 (2's).

Cf. A283316 (mod 2).

Programs

PARI

a(n) = #select(x->(x==3), digits(n, 4)); \\ Michel Marcus, Mar 24 2020

Formula

Recurrence relation: a(0) = 0, a(4m+3) = 1+a(m), a(4m) = a(4m+1) = a(4m+2) = a(m).

G.f.: (1/(1-z))*Sum_{m>=0} (z^(3*4^m)*(1 - z^(4^m))/(1 - z^(4^(m+1)))).

Morphism: 0, j -> j,j,j,j+1; e.g., 0 -> 0001 -> 0001000100011112 -> ...