A047495 - OEIS

%I #18 Sep 08 2022 08:44:57

%S 2,4,5,7,10,12,13,15,18,20,21,23,26,28,29,31,34,36,37,39,42,44,45,47,

%T 50,52,53,55,58,60,61,63,66,68,69,71,74,76,77,79,82,84,85,87,90,92,93,

%U 95,98,100,101,103,106,108,109,111,114,116,117,119,122,124

%N Numbers that are congruent to {2, 4, 5, 7} mod 8.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F G.f.: x*(2+x^2+x^3) / ( (x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Nov 06 2015

%F From _Wesley Ivan Hurt_, May 27 2016: (Start)

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

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

%F a(2k) = A047535(k), a(2k-1) = A047617(k). (End)

%F E.g.f.: (2 + sin(x) - cos(x) + (4*x - 1)*exp(x))/2. - _Ilya Gutkovskiy_, May 27 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 - (sqrt(2)-1)*log(2)/8 + sqrt(2)*log(2-sqrt(2))/4. - _Amiram Eldar_, Dec 25 2021

%p A047495:=n->(1+I)*(4*n-4*n*I+I-1+I^(1-n)-I^n)/4: seq(A047495(n), n=1..100); # _Wesley Ivan Hurt_, May 27 2016

%t Table[(1+I)*(4n-4n*I+I-1+I^(1-n)-I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 27 2016 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [2, 4, 5, 7]]; // _Wesley Ivan Hurt_, May 27 2016

%Y Cf. A047535, A047617.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_