A047492 - OEIS

%I #20 Sep 08 2022 08:44:57

%S 0,4,5,7,8,12,13,15,16,20,21,23,24,28,29,31,32,36,37,39,40,44,45,47,

%T 48,52,53,55,56,60,61,63,64,68,69,71,72,76,77,79,80,84,85,87,88,92,93,

%U 95,96,100,101,103,104,108,109,111,112,116,117,119,120,124

%N Numbers that are congruent to {0, 4, 5, 7} mod 8.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F G.f.: x^2*(4+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - _R. J. Mathar_, Nov 06 2015

%F From _Wesley Ivan Hurt_, May 26 2016: (Start)

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

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

%F a(2k) = A047535(k), a(2k-1) = A047615(k). (End)

%F E.g.f.: (2 - sin(x) - cos(x) + (4*x - 3)*sinh(x) + (4*x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 27 2016

%F Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4 - Pi/8. - _Amiram Eldar_, Dec 23 2021

%p A047492:=n->2*n+(1+I)*(2*I-2+(1-I)*I^(2*n)-I^(-n)+I^(1+n))/4: seq(A047492(n), n=1..100); # _Wesley Ivan Hurt_, May 26 2016

%t Table[2n+(1+I)*(2*I-2+(1-I)*I^(2n)-I^(-n)+I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, May 26 2016 *)

%t a[n_] := 1 + n + Floor[n/2] + 2 Floor[(n - 2)/4];

%t Table[a[n], {n, 1, 62}] (* _Peter Luschny_, Dec 23 2021 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [0, 4, 5, 7]]; // _Wesley Ivan Hurt_, May 26 2016

%Y Cf. A047535, A047615.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_