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A047431
Numbers that are congruent to {1, 4, 5, 6} mod 8.
5
1, 4, 5, 6, 9, 12, 13, 14, 17, 20, 21, 22, 25, 28, 29, 30, 33, 36, 37, 38, 41, 44, 45, 46, 49, 52, 53, 54, 57, 60, 61, 62, 65, 68, 69, 70, 73, 76, 77, 78, 81, 84, 85, 86, 89, 92, 93, 94, 97, 100, 101, 102, 105, 108, 109, 110, 113, 116, 117, 118, 121, 124
OFFSET
1,2
FORMULA
G.f.: x*(1+2*x-x^2+2*x^3)/((x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = (-2-(-i)^n-i^n+4n)/2 where i=sqrt(-1). - Colin Barker, Jun 06 2012
From Wesley Ivan Hurt, May 30 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047406(k), a(2k-1) = A016813(k-1) k>0. (End)
E.g.f.: 2 - cos(x) - (1 - 2*x)*exp(x). - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 24 2021
MAPLE
A047431:=n->(4*n-2-(-I)^n-I^n)/2: seq(A047431(n), n=1..100); # Wesley Ivan Hurt, May 30 2016
MATHEMATICA
Table[(4n-2-(-I)^n-I^n)/2, {n, 80}] (* Wesley Ivan Hurt, May 30 2016 *)
LinearRecurrence[{2, -2, 2, -1}, {1, 4, 5, 6}, 70] (* Harvey P. Dale, Dec 04 2018 *)
PROG
(Sage) [lucas_number1(n, 0, 1)+2*n+1 for n in range(0, 56)] # Zerinvary Lajos, Jul 06 2008
(Magma) [n : n in [0..150] | n mod 8 in [1, 4, 5, 6]]; // Wesley Ivan Hurt, May 30 2016
CROSSREFS
KEYWORD
nonn,easy
STATUS
approved