A047612 Numbers that are congruent to {0, 2, 4, 5} mod 8.
0, 2, 4, 5, 8, 10, 12, 13, 16, 18, 20, 21, 24, 26, 28, 29, 32, 34, 36, 37, 40, 42, 44, 45, 48, 50, 52, 53, 56, 58, 60, 61, 64, 66, 68, 69, 72, 74, 76, 77, 80, 82, 84, 85, 88, 90, 92, 93, 96, 98, 100, 101, 104, 106, 108, 109, 112, 114, 116, 117, 120, 122, 124
Offset: 1
Links
- Bruno Berselli, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Programs
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Magma
[n: n in [0..120] | n mod 8 in [0,2,4,5]];
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Maple
A047612:=n->2*n-2-(1+I^(2*n))*(1+I^n)/4: seq(A047612(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016
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Mathematica
Select[Range[0,120], MemberQ[{0, 2, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {0, 2, 4, 5, 8}, 60] (* Bruno Berselli, Jul 18 2012 *) -
Maxima
makelist(2*n-2-(1+(-1)^n)*(1+%i^n)/4,n,1,60);
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PARI
concat(0, Vec((2+2*x+x^2+3*x^3)/((1+x)*(1-x)^2*(1+x^2))+O(x^60))) (End)
Formula
From Bruno Berselli, Jul 18 2012: (Start)
G.f.: x^2*(2+2*x+x^2+3*x^3)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-2-(1+(-1)^n)*(1+i^n)/4, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
E.g.f.: (6 - cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + 5*log(2)/8 + sqrt(2)*log(sqrt(2)-1)/8. - Amiram Eldar, Dec 21 2021