A047585 Numbers that are congruent to {0, 1, 3, 5, 6, 7} mod 8.

0, 1, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 35, 37, 38, 39, 40, 41, 43, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 85, 86, 87, 88

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Formula

From Chai Wah Wu, Jun 10 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6).

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

a(n) = (12*n - 3*sqrt(3)*sin(Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) - 9)/9. - Ilya Gutkovskiy, Jun 11 2016

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

Sum_{n>=2} (-1)^n/a(n) = (3-2*sqrt(2))*Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 27 2021

Maple

A047585:=n->(12*n - 3*sqrt(3)*sin(Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) - 9)/9: seq(A047585(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

Select[Range[0, 100], MemberQ[{0, 1, 3, 5, 6, 7}, Mod[#, 8]]&] (* or *) Complement[Range[0, 100], Flatten[Range[{2, 4}, 100, 8]]] (* Harvey P. Dale, May 01 2012 *)
CoefficientList[Series[x (x^4 + x^2 + x + 1) / ((x - 1)^2 (x^2 - x + 1) (x^2 + x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 18 2016 *)

Prog

(Magma) [n : n in [0..100] | n mod 8 in [0, 1, 3, 5, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016

Crossrefs

Cf. A047517, A047569.

Sequence in context: A289178 A212450 A368828 * A288224 A039063 A266114

Adjacent sequences: A047582 A047583 A047584 * A047586 A047587 A047588

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 11 1999

Status

approved