A047418 Numbers that are congruent to {0, 2, 3, 4, 6} mod 8.
0, 2, 3, 4, 6, 8, 10, 11, 12, 14, 16, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 40, 42, 43, 44, 46, 48, 50, 51, 52, 54, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 72, 74, 75, 76, 78, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
Programs
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GAP
Filtered([0..103],n->n mod 8 = 0 or n mod 8 = 2 or n mod 8 = 3 or n mod 8 = 4 or n mod 8 = 6); # Muniru A Asiru, Oct 23 2018
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Magma
[n : n in [0..150] | n mod 8 in [0, 2, 3, 4, 6]]; // Wesley Ivan Hurt, Aug 08 2016
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Maple
A047418:=n->8*floor(n/5)+[(0, 2, 3, 4, 6)][(n mod 5)+1]: seq(A047418(n), n=0..100); # Wesley Ivan Hurt, Aug 08 2016
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Mathematica
Select[Range[0,100], MemberQ[{0,2,3,4,6}, Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,0,0,1,-1}, {0,2,3,4,6,8}, 70] (* Harvey P. Dale, Oct 01 2015 *)
Formula
G.f.: x^2*(2 + x + x^2 + 2*x^3 + 2*x^4)/((x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, Aug 08 2016: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6, a(n) = a(n-5) + 8 for n > 5.
a(n) = (40*n - 45 - 2*(n mod 5) + 3*((n + 1) mod 5) + 3*((n + 2) mod 5) - 2*((n + 3) mod 5) - 2*((n + 4) mod 5))/25.
a(5*k) = 8*k - 2, a(5*k-1) = 8*k - 4, a(5*k-2) = 8*k - 5, a(5*k-3) = 8*k - 6, a(5*k-4) = 8*k - 8. (End)
a(n) = (40*n - 45 + 2*cos(2*Pi*(n - 1)/5) - 2*cos(2*Pi*n/5) - 2*cos(4*Pi*n/5) - 6*cos(2*Pi*(n + 1)/5) - 6*cos(Pi*(2*n + 1)/5) + 6*cos(2*Pi*(2*n + 1)/5) - 2*cos(Pi*(4*n + 1)/5) + 6*sin(Pi*(8*n + 3)/10))/25. - Wesley Ivan Hurt, Oct 10 2018