A047424 Numbers that are congruent to {0, 2, 3, 4, 5, 6} mod 8.

0, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 32, 34, 35, 36, 37, 38, 40, 42, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Crossrefs

Cf. A047428, A047503.

Programs

Magma

[n : n in [0..100] | n mod 8 in [0] cat [2..6]]; // Wesley Ivan Hurt, Jun 15 2016

Maple

A047424:=n->(12*n-12-sqrt(3)*cos((1-4*n)*Pi/6)-3*sqrt(3)*cos((1+2*n)*Pi/6))/9: seq(A047424(n), n=1..100); # Wesley Ivan Hurt, Jun 15 2016

Mathematica

Select[Range[0,100], MemberQ[{0,2,3,4,5,6}, Mod[#,8]]&] (* Harvey P. Dale, Mar 21 2011 *)

PARI

my(x='x+O('x^50)); concat([0], Vec(x^2*(2 -x +2*x^2 -x^3 +2*x^4 )/((1 + x+x^2)*(x^2-x+1)*(x-1)^2))) \\ G. C. Greubel, Oct 29 2017

Formula

G.f.: x^2*(2-x+2*x^2-x^3+2*x^4) / ( (1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, Jun 15 2016: (Start)

a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-a(n-6) for n>6.

a(n) = (12*n-12-sqrt(3)*cos((1-4*n)*Pi/6)-3*sqrt(3)*cos((1+2*n)*Pi/6))/9.

a(6k) = 8k-2, a(6k-1) = 8k-3, a(6k-2) = 8k-4, a(6k-3) = 8k-5, a(6k-4) = 8k-6, a(6k-5) = 8k-8. (End)

Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*log(3+2*sqrt(2))/8. - Amiram Eldar, Dec 27 2021