A047401 Numbers that are congruent to {0, 1, 3, 6} mod 8.

0, 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 22, 24, 25, 27, 30, 32, 33, 35, 38, 40, 41, 43, 46, 48, 49, 51, 54, 56, 57, 59, 62, 64, 65, 67, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 97, 99, 102, 104, 105, 107, 110, 112, 113, 115, 118, 120, 121, 123

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Partial sums of A068073. - Jeremy Gardiner, Jul 20 2013.

Links

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

Crossrefs

Cf. A047452, A047470, A068073.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 1, 3, 6]]; // Wesley Ivan Hurt, Jun 01 2016

Maple

A047401:=n->2*(n-1)+(I^(n*(n-1))-1)/2: seq(A047401(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Mathematica

Select[Range[0,107], MemberQ[{0, 1, 3, 6}, Mod[#, 8]]&] (* Bruno Berselli, Dec 05 2011 *)

Maxima

makelist(2*(n-1)+(%i^(n*(n-1))-1)/2,n,1,55); /* Bruno Berselli, Dec 05 2011 */

PARI

my(x='x+O('x^100)); concat(0, Vec(x^2*(1+x+2*x^2)/((x^2+1)*(x-1)^2))) \\ Altug Alkan, Jun 02 2016

Formula

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

a(n) = 2*(n-1)+(i^(n*(n-1))-1)/2, where i=sqrt(-1). - Bruno Berselli, Dec 05 2011

From Wesley Ivan Hurt, Jun 01 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(2k) = A047452(k), a(2k-1) = A047470(k). (End)

Sum_{n>=2} (-1)^n/a(n) = Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021