A047569 Numbers that are congruent to {0, 1, 4, 5, 6, 7} mod 8.

0, 1, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A047517, A047585.

Programs

Maple

A047569:=n->(24*n-15-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/18: seq(A047569(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

Select[Range[0, 100], MemberQ[{0, 1, 4, 5, 6, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)
LinearRecurrence[{1,0,0,0,0,1,-1},{0,1,4,5,6,7,8},80] (* Harvey P. Dale, Feb 15 2024 *)

PARI

concat(0, Vec(x^2*(1+3*x+x^2+x^3+x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 09 2016

Formula

G.f.: x^2*(1+3*x+x^2+x^3+x^4+x^5) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)). - Colin Barker, Jan 09 2016

From Wesley Ivan Hurt, Jun 16 2016: (Start)

a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.

a(n) = (24*n-15-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/18.

a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-3, a(6k-3) = 8k-4, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)

Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + (14-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 27 2021