A047503 Numbers that are congruent to {0, 2, 3, 4, 5, 7} mod 8.

0, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 39, 40, 42, 43, 44, 45, 47, 48, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 63, 64, 66, 67, 68, 69, 71, 72, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 87

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A047424, A047587.

Programs

Magma

[n : n in [0..100] | n mod 8 in [0, 2, 3, 4, 5, 7]]; // Wesley Ivan Hurt, Jun 16 2016

Maple

A047503:=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(A047503(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

Select[Range[0, 100], MemberQ[{0, 2, 3, 4, 5, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)
LinearRecurrence[{1,0,0,0,0,1,-1},{0,2,3,4,5,7,8},100] (* Harvey P. Dale, Dec 25 2023 *)

Formula

G.f.: x^2*(2+x+x^2+x^3+2*x^4+x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015

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

Sum_{n>=2} (-1)^n/a(n) = log(2)/8 - sqrt(2)*Pi/16 - sqrt(2)*log(99-70*sqrt(2))/16. - Amiram Eldar, Dec 27 2021