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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

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

Programs

Magma

[n : n in [0..150] | n mod 8 in [1, 4, 5, 6]]; // Wesley Ivan Hurt, May 30 2016

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 *)

Sage

[lucas_number1(n,0,1)+2*n+1 for n in range(0,56)] # Zerinvary Lajos, Jul 06 2008

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