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