A047564 Numbers that are congruent to {1, 3, 4, 5, 6, 7} mod 8.
1, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 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 (2,-2,2,-2,2,-1).
Programs
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Magma
[n : n in [0..100] | n mod 8 in [1, 3, 4, 5, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016
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Maple
A047564:=n->(12*n-3-sqrt(3)*(cos((1-4*n)*Pi/6)+3*cos((1+2*n)*Pi/6)))/9: seq(A047564(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[0, 100], MemberQ[{1, 3, 4, 5, 6, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *) CoefficientList[Series[(x^5 + x^3 + 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, May 30 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) for n>6.
G.f.: x*(x^5 + x^3 + x + 1)/((x - 1)^2*(x^2 - x + 1)*(x^2 + x + 1)). (End)
From Wesley Ivan Hurt, Jun 16 2016: (Start)
a(n) = (12*n-3-sqrt(3)*(cos((1-4*n)*Pi/6)+3*cos((1+2*n)*Pi/6)))/9.
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-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/4 - (sqrt(2)+3)*log(2)/8. - Amiram Eldar, Dec 28 2021