A047623 Numbers that are congruent to {1, 3, 5} mod 8.

1, 3, 5, 9, 11, 13, 17, 19, 21, 25, 27, 29, 33, 35, 37, 41, 43, 45, 49, 51, 53, 57, 59, 61, 65, 67, 69, 73, 75, 77, 81, 83, 85, 89, 91, 93, 97, 99, 101, 105, 107, 109, 113, 115, 117, 121, 123, 125, 129, 131, 133, 137, 139, 141, 145, 147, 149, 153, 155, 157

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers that can be expressed as the sum of at most three square numbers (see Tattersall). - Stefano Spezia, Jul 02 2025

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 17.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A047478, A047484, A047529.

Programs

Magma

I:=[1, 3, 5, 9]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 27 2012

Maple

A047623:=n->(24*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047623(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Mathematica

Select[Range[0,150], MemberQ[{1,3,5}, Mod[#,8]]&] (* Vincenzo Librandi, Apr 27 2012 *)

Formula

a(n) = 2*floor((n-1)/3) + 2*n - 1. - Gary Detlefs, Mar 18 2010

From Colin Barker, Feb 03 2012: (Start)

G.f.: x*(1+2*x+2*x^2+3*x^3)/(1-x-x^3+x^4).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (End)

From Wesley Ivan Hurt, Jun 10 2016: (Start)

a(n) = (24*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 8k-3, a(3k-1) = 8k-5, a(3k-2) = 8k-7. (End)