A047411 Numbers that are congruent to {1, 2, 4, 6} mod 8.

1, 2, 4, 6, 9, 10, 12, 14, 17, 18, 20, 22, 25, 26, 28, 30, 33, 34, 36, 38, 41, 42, 44, 46, 49, 50, 52, 54, 57, 58, 60, 62, 65, 66, 68, 70, 73, 74, 76, 78, 81, 82, 84, 86, 89, 90, 92, 94, 97, 98, 100, 102, 105, 106, 108, 110, 113, 114, 116, 118, 121, 122, 124, 126, 129, 130

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A016825, A047461.

Programs

Magma

[n : n in [0..150] | n mod 8 in [1, 2, 4, 6]]; // Wesley Ivan Hurt, May 22 2016

Maple

A047411 := proc(n) if n <= 4 then op(n,[1,2,4,6]); else procname(n-4)+8; end if; end proc: seq(A047411(n), n=1..99); # R. J. Mathar, Feb 11 2010

Mathematica

Table[(8n-7-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{1,0,0,1,-1}, {1,2,4,6,9}, 50] (* G. C. Greubel, May 23 2016 *)

Formula

From R. J. Mathar, Mar 10 2008: (Start)

a(n) = a(n-4)+8.

O.g.f.: 2/(-1+x)^2+1/(2*(x^2+1))+7/(4*(-1+x))+1/(4*(x+1)). (End)

a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5. - R. J. Mathar, Feb 11 2010

From Wesley Ivan Hurt, May 22 2016: (Start)

a(n) = (8*n-7-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).

a(2*n+2) = A016825(n) n>0, a(2*n-1) = A047461(n). (End)

E.g.f.: (4 + sin(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 23 2016

a(n) = (8*n-7-cos(n*Pi)+2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 05 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 23 2021

Extensions

Extended by R. J. Mathar, Feb 11 2010