A039301 Number of distinct quadratic residues mod 4^n.

1, 2, 4, 12, 44, 172, 684, 2732, 10924, 43692, 174764, 699052, 2796204, 11184812, 44739244, 178956972, 715827884, 2863311532, 11453246124, 45812984492, 183251937964, 733007751852, 2932031007404, 11728124029612, 46912496118444, 187649984473772, 750599937895084

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Number of distinct n-digit suffixes of base 4 squares.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..170
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Programs

Magma

[Floor((4^n+10)/6): n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

Mathematica

Floor[(4^Range[0, 30] + 10)/6] (* Paolo Xausa, Jan 29 2025 *)

Formula

a(n) = floor((4^n+10)/6).

a(n) = A007583(n-1)+1 = A020988(n-2)+2 = A083584(n-2)+3. - Ralf Stephan, Jun 14 2003

Also, a(0)=1 and, for n>0, a(n) = (4^n+8)/6. - Bruno Berselli, Jul 27 2010

G.f.: (1-3*x-2*x^2)/((1-x)*(1-4*x)). - Bruno Berselli, Jul 27 2010

a(n)-5*a(n-1)+4*a(n-2) = 0 for n>1. - Bruno Berselli, Jul 27 2010