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

%I A047439 #28 Sep 08 2022 08:44:57
%S A047439 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,
%T A047439 49,53,54,56,57,61,62,64,65,69,70,72,73,77,78,80,81,85,86,88,89,93,94,
%U A047439 96,97,101,102,104,105,109,110,112,113,117,118,120,121,125
%N A047439 Numbers that are congruent to {0, 1, 5, 6} mod 8.
%H A047439 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047439 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
%F A047439 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
%F A047439 a(n) = Sum_{i=1..n} gcd(i+2, i-2). - _Wesley Ivan Hurt_, Jan 23 2014
%F A047439 From _Wesley Ivan Hurt_, May 22 2016: (Start)
%F A047439 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047439 a(n) = 2n+(1+i)*(4i-4-(1-i)*i^(2n)-i^(1-n)+i^n)/4 where i=sqrt(-1).
%F A047439 a(2n) = A047452, a(2n-1) = A047615(n). (End)
%F A047439 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
%p A047439 A047439:=n->add(gcd(i+2, i-2), i=1..n); seq(A047439(n), n=0..100); # _Wesley Ivan Hurt_, Jan 23 2014
%t A047439 Table[Sum[GCD[i + 2, i - 2], {i, n}], {n, 0, 100}] (* _Wesley Ivan Hurt_, Jan 23 2014 *)
%o A047439 (Magma) [n : n in [0..150] | n mod 8 in [0, 1, 5, 6]]; // _Wesley Ivan Hurt_, May 22 2016
%Y A047439 Cf. A030308, A047452, A047615.
%K A047439 nonn,easy
%O A047439 1,3
%A A047439 _N. J. A. Sloane_
%E A047439 More terms from _Wesley Ivan Hurt_, May 22 2016