A047428 Numbers that are congruent to {0, 1, 3, 4, 5, 6} mod 8.
0, 1, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 25, 27, 28, 29, 30, 32, 33, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 48, 49, 51, 52, 53, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 88
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, 1, 3, 4, 5, 6]]; // Wesley Ivan Hurt, Jun 16 2016
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Maple
A047428:=n->(24*n-27-3*cos(n*Pi)-6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18: seq(A047428(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[0, 100], MemberQ[{0, 1, 3, 4, 5, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)
Formula
G.f.: x^2*(1+2*x+x^2+x^3+x^4+2*x^5) / ((1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2). - R. J. Mathar, Dec 07 2011
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-27-3*cos(n*Pi)-6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18.
a(6k) = 8k-2, a(6k-1) = 8k-3, a(6k-2) = 8k-4, a(6k-3) = 8k-5, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*Pi/16 + log(2)/8 - sqrt(2)*log(99-70*sqrt(2))/16. - Amiram Eldar, Dec 27 2021