A047581 Numbers that are congruent to {0, 1, 2, 5, 6, 7} mod 8.

0, 1, 2, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88

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. A047498, A047513.

Programs

Magma

[n : n in [0..100] | n mod 8 in [0, 1, 2, 5, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016

Maple

A047581:=n->(8*n+(-1)^n-2*sqrt(3)*sin(Pi*n/3)-4*sin(2*Pi*(n+1)/3)/sqrt(3)
+2*cos(Pi*n/3)-7)/6: seq(A047581(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 5, 6, 7, 8}, 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^2*(x^5 + x^4 + x^3 + 3*x^2 + x + 1)/(x^7 - x^6 - x + 1). (End)

a(n) = (8*n + (-1)^n - 2*sqrt(3)*sin(Pi*n/3) - 4*sin(2*Pi*(n+1)/3)/sqrt(3) + 2*cos(Pi*n/3) - 7)/6. - Ilya Gutkovskiy, May 30 2016

a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-3, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. - Wesley Ivan Hurt, Jun 16 2016

Sum_{n>=2} (-1)^n/a(n) = (12-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 27 2021