A047407 Numbers that are congruent to {0, 4, 6} mod 8.

0, 4, 6, 8, 12, 14, 16, 20, 22, 24, 28, 30, 32, 36, 38, 40, 44, 46, 48, 52, 54, 56, 60, 62, 64, 68, 70, 72, 76, 78, 80, 84, 86, 88, 92, 94, 96, 100, 102, 104, 108, 110, 112, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 142, 144, 148, 150, 152, 156

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A004772, A047395, A047410, A047464.

Programs

Magma

[n : n in [0..160] | n mod 8 in [0, 4, 6]]; // Vincenzo Librandi, May 02 2016

Maple

A047407:=n->2*(12*n-9-2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047407(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Mathematica

Select[Range[0,200], MemberQ[{0,4,6}, Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,1,-1}, {0,4,6,8}, 70] (* Harvey P. Dale, Apr 20 2016 *)

PARI

a(n)=n\3*8+[-2,0,4][n%3+1] \\ Charles R Greathouse IV, May 02 2016

Formula

From R. J. Mathar, Dec 05 2011: (Start)

a(n) = 2*A004772(n).

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

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

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

a(n) = 2*(12*n-9-2*sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 8k-2, a(3k-1) = 8k-4, a(3k-2) = 8k-8. (End)

a(n) = 2*(n - 1 + floor((n + 1)/3)). - Wolfdieter Lang, Sep 11 2021

Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 19 2021