A047439 Numbers that are congruent to {0, 1, 5, 6} mod 8.

0, 1, 5, 6, 8, 9, 13, 14, 16, 17, 21, 22, 24, 25, 29, 30, 32, 33, 37, 38, 40, 41, 45, 46, 48, 49, 53, 54, 56, 57, 61, 62, 64, 65, 69, 70, 72, 73, 77, 78, 80, 81, 85, 86, 88, 89, 93, 94, 96, 97, 101, 102, 104, 105, 109, 110, 112, 113, 117, 118, 120, 121, 125

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. A030308, A047452, A047615.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 1, 5, 6]]; // Wesley Ivan Hurt, May 22 2016

Maple

A047439:=n->add(gcd(i+2, i-2), i=1..n); seq(A047439(n), n=0..100); # Wesley Ivan Hurt, Jan 23 2014

Mathematica

Table[Sum[GCD[i + 2, i - 2], {i, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 23 2014 *)

Formula

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=5 and b(k)=2^(k+1) for k>1. - Philippe Deléham, Oct 19 2011

G.f.: x^2*(1+4*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011

a(n) = Sum_{i=1..n} gcd(i+2, i-2). - Wesley Ivan Hurt, Jan 23 2014

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

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

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

a(2n) = A047452, a(2n-1) = A047615(n). (End)

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

Extensions

More terms from Wesley Ivan Hurt, May 22 2016