A047498 Numbers that are congruent to {0, 1, 2, 4, 5, 7} mod 8.
0, 1, 2, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 28, 29, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 45, 47, 48, 49, 50, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 84, 85, 87, 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, 2, 4, 5, 7]]; // Wesley Ivan Hurt, Jun 16 2016
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Maple
A047498:=n->(24*n-27+3*cos(n*Pi)+6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18: seq(A047498(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[0,100],MemberQ[{0,1,2,4,5,7},Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,0,0,0,1,-1},{0,1,2,4,5,7,8},100] (* Harvey P. Dale, Jul 23 2015 *)
Formula
G.f.: x^2*(x^5+2*x^4+x^3+2*x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jun 22 2012
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-1, a(6k-1) = 8k-3, a(6k-2) = 8k-4, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (6-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8 - (2-sqrt(2))*Pi/16. - Amiram Eldar, Dec 27 2021