A047593 Numbers that are congruent to {2, 3, 4, 5, 6, 7} mod 8.
2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87
Offset: 1
Links
- Vincenzo Librandi, 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).
Programs
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Magma
[n: n in [1..80] | n mod 8 in [2..7]]; // Vincenzo Librandi, Jan 06 2013
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Maple
A047593:=n->(24*n-3-3*cos(n*Pi)-4*sqrt(3)*cos((1+4*n)*Pi/6)-12*sin((1-2*n)*Pi/6))/18: seq(A047593(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[100], MemberQ[{2, 3, 4, 5, 6, 7}, Mod[#, 8]]&] (* Vincenzo Librandi, Jan 06 2013 *)
Formula
G.f.: x*(2+x+x^2+x^3+x^4+x^5+x^6) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Jul 10 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-3-3*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-5, a(6k-5) = 8k-6. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)+8)*log(2)/16. - Amiram Eldar, Dec 28 2021