A047572 Numbers that are congruent to {1, 2, 4, 5, 6, 7} mod 8.
1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 81, 82, 84, 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 [1, 2, 4, 5, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016
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Maple
A047572:=n->(24*n-9-3*cos(n*Pi)-6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18: seq(A047572(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[0, 100], MemberQ[{1, 2, 4, 5, 6, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *) -
PARI
a(n)=n\6*8+[-1,1,2,4,5,6][n%6+1] \\ Charles R Greathouse IV, Feb 24 2015
Formula
From Wesley Ivan Hurt, Jun 16 2016: (Start)
G.f.: x*(1+x+2*x^2+x^3+x^4+x^5+x^6) / ((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-9-3*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-2, a(6k-2) = 8k-3, a(6k-3) = 8k-4, a(6k-4) = 8k-6, a(6k-5) = 8k-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)+4)*log(2)/16. - Amiram Eldar, Dec 28 2021