A047403 Numbers that are congruent to {0, 2, 3, 6} mod 8.

0, 2, 3, 6, 8, 10, 11, 14, 16, 18, 19, 22, 24, 26, 27, 30, 32, 34, 35, 38, 40, 42, 43, 46, 48, 50, 51, 54, 56, 58, 59, 62, 64, 66, 67, 70, 72, 74, 75, 78, 80, 82, 83, 86, 88, 90, 91, 94, 96, 98, 99, 102, 104, 106, 107, 110, 112, 114, 115, 118, 120, 122, 123, 126, 128

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A016825, A047470.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 2, 3, 6]]; // Wesley Ivan Hurt, May 24 2016

Maple

A047403:=n->(8*n-9+I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047403(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Mathematica

Table[(8n-9+I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 24 2016 *)
#+{0,2,3,6}&/@(8*Range[0,20])//Flatten (* or *) LinearRecurrence[{1,0,0,1,-1},{0,2,3,6,8},80] (* Harvey P. Dale, Mar 02 2023 *)

Formula

a(n) = 2*n - ((n mod 4) == 2).

G.f.: x^2*(2+x+3*x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011

From Wesley Ivan Hurt, May 24 2016: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

a(n) = (8*n-9+i^(2*n)+i^(1-n)-i^(1+n))/4, where i=sqrt(-1).

a(2k) = A016825(k-1) for k>0, a(2k-1) = A047470. (End)

E.g.f.: (4 + sin(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016

Sum_{n>=2} (-1)^n/a(n) = (4-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 21 2021