A047569 Numbers that are congruent to {0, 1, 4, 5, 6, 7} mod 8.

0, 1, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88

Offset

1,3

Links

Colin Barker, 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

G.f.: x^2*(1+3*x+x^2+x^3+x^4+x^5) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)). - Colin Barker, Jan 09 2016

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

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

a(n) = (24*n-15-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/18.

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

Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + (14-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 27 2021

Maple

A047569:=n->(24*n-15-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/18: seq(A047569(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

Select[Range[0, 100], MemberQ[{0, 1, 4, 5, 6, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 4, 5, 6, 7, 8}, 80] (* Harvey P. Dale, Feb 15 2024 *)

Prog

(PARI) concat(0, Vec(x^2*(1+3*x+x^2+x^3+x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 09 2016

Crossrefs

Cf. A047517, A047585.

Sequence in context: A016070 A299536 A321025 * A039062 A067187 A321353

Adjacent sequences: A047566 A047567 A047568 * A047570 A047571 A047572

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved