A047517 Numbers that are congruent to {0, 1, 3, 4, 6, 7} mod 8.

0, 1, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 20, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 86, 87, 88, 89

Offset

1,3

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).

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 + 2*x^3 + x^2 + 2*x + 1)/(x^7 - x^6 - x + 1). (End)

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

a(n) = (24*n-21-3*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-7, a(6k-5) = 8k-8. (End)

Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + (6-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 27 2021

Maple

A047424:=n->(24*n-21-3*cos(n*Pi)+2*sqrt(3)*cos((1+4*n)*Pi/6)+6*sin((1-2*n)* Pi/6))/18: seq(A047424(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 4, 6, 7, 8}, 50] (* G. C. Greubel, May 30 2016 *)
Select[Range[0, 200], MemberQ[{0, 1, 3, 4, 6, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, May 30 2016 *)

Prog

(Magma) [n: n in [0..110] | n mod 8 in [0, 1, 3, 4, 6, 7]]; // Vincenzo Librandi, May 30 2016
(PARI) my(x='x+O('x^50)); concat([0], Vec(x^2*(x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^7 - x^6 - x + 1))) \\ G. C. Greubel, Oct 29 2017

Crossrefs

Cf. A047428, A047585.

Sequence in context: A182835 A058654 A188435 * A206587 A192450 A039064

Adjacent sequences: A047514 A047515 A047516 * A047518 A047519 A047520

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved