A047431 Numbers that are congruent to {1, 4, 5, 6} mod 8.

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

Links

Table of n, a(n) for n=1..62.

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

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

Cf. A016813, A047404, A047406, A047546, A056594.

Sequence in context: A010456 A072496 A306482 * A288695 A182719 A003156

Adjacent sequences: A047428 A047429 A047430 * A047432 A047433 A047434

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved