A047587 Numbers that are congruent to {0, 2, 3, 5, 6, 7} mod 8.
0, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 69, 70, 71, 72, 74, 75, 77, 78, 79, 80, 82, 83, 85, 86, 87
Offset: 1
Links
- 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 [0..100] | n mod 8 in [0, 2, 3, 5, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016
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Maple
A047587:=n->(24*n-15+3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)-6*sin((1+2*n)*Pi/6))/18: seq(A047587(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[0,150], MemberQ[{0,2,3,5,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Oct 04 2011 *)
Formula
From Wesley Ivan Hurt, Jun 16 2016: (Start)
G.f.: x^2*(2+x+2*x^2+x^3+x^4+x^5)/((x-1)^2*(1+x+x^2+x^3+x^4+x^5)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = (24*n-15+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-2, a(6k-2) = 8k-3, a(6k-3) = 8k-5, a(6k-4) = 8k-6, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (8-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - 3*(sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 27 2021