A047563 Numbers that are congruent to {0, 3, 4, 5, 6, 7} mod 8.

0, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87

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 (1,0,0,0,0,1,-1).

Crossrefs

Cf. A047571.

Programs

Magma

[n : n in [0..100] | n mod 8 in [0] cat [3..7]]; // Wesley Ivan Hurt, May 29 2016

Maple

A047563:=n->(24*n-9+3*cos(n*Pi)-12*cos(n*Pi/3)-4*sqrt(3)*sin(2*n*Pi/3))/18: seq(A047563(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 3, 4, 5, 6, 7, 8}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

From Chai Wah Wu, May 29 2016: (Start)

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

G.f.: x^2*(x^5 + x^4 + x^3 + x^2 + x + 3)/(x^7 - x^6 - x + 1). (End)

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

a(n) = (24*n-9+3*cos(n*Pi)-12*cos(n*Pi/3)-4*sqrt(3)*sin(2*n*Pi/3))/18.

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

Sum_{n>=2} (-1)^n/a(n) = 7*log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16 - sqrt(2)*Pi/16. - Amiram Eldar, Dec 27 2021