A047519 Numbers that are congruent to {1, 2, 3, 4, 6, 7} mod 8.

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 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. A047422, A047504.

Programs

Magma

[n : n in [0..100] | n mod 8 in [1, 2, 3, 4, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016

Maple

A047519:=n->(24*n-15-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n)
*Pi/6))/18: seq(A047519(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {1, 2, 3, 4, 6, 7, 9}, 50] (* G. C. Greubel, May 30 2016 *)

Formula

From Chai Wah Wu, May 30 2016: (Start)

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

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

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

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

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

Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-1)*Pi/16 + log(2)/4 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 28 2021