A047605 Numbers that are congruent to {0, 2, 3, 5} mod 8.
0, 2, 3, 5, 8, 10, 11, 13, 16, 18, 19, 21, 24, 26, 27, 29, 32, 34, 35, 37, 40, 42, 43, 45, 48, 50, 51, 53, 56, 58, 59, 61, 64, 66, 67, 69, 72, 74, 75, 77, 80, 82, 83, 85, 88, 90, 91, 93, 96, 98, 99, 101, 104, 106, 107, 109, 112, 114, 115, 117, 120, 122, 123
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Programs
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Magma
[n : n in [0..150] | n mod 8 in [0, 2, 3, 5]]; // Wesley Ivan Hurt, Jun 04 2016
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Maple
A047605:=n->2*(n-1)-(I^(n*(n+1))+1)/2: seq(A047605(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016
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Mathematica
Table[(1+I)*((4-4*I)*n+5*I-5+I^(1-n)-I^n)/4, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *) Flatten[#+{0,2,3,5}&/@(8*Range[0,20])] (* or *) LinearRecurrence[{2,-2,2,-1},{0,2,3,5},100] (* Harvey P. Dale, Sep 30 2018 *) -
PARI
a(n)=n\4*8+[-3,0,2,3][n%4+1] \\ Charles R Greathouse IV, Dec 05 2011
Formula
From Bruno Berselli, Dec 05 2011: (Start)
G.f.: x^2*(2-x+3*x^2)/((1-x)^2*(1+x^2)).
a(n) = 2*(n-1)-(i^(n*(n+1))+1)/2, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
E.g.f.: (6 + sin(x) - cos(x) + (4*x - 5)*exp(x))/2. - Ilya Gutkovskiy, Jun 05 2016
Sum_{n>=2} (-1)^n/a(n) = (3-2*sqrt(2))*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 21 2021