|
%I #67 Aug 03 2022 02:35:59
%S 0,6,8,14,16,22,24,30,32,38,40,46,48,54,56,62,64,70,72,78,80,86,88,94,
%T 96,102,104,110,112,118,120,126,128,134,136,142,144,150,152,158,160,
%U 166,168,174,176,182,184,190,192,198,200,206,208,214,216,222,224,230
%N Numbers that are congruent to {0, 6} mod 8.
%C All even numbers m such that Integral_{x=0..2*Pi} Product_{i=1..m/2} cos(2*i*x) dx is nonzero. - _William Boyles_, Oct 12 2019
%H Michael De Vlieger, <a href="/A047451/b047451.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = 8*n - a(n-1) - 10 (with a(1)=0). - _Vincenzo Librandi_, Aug 06 2010
%F From _R. J. Mathar_, Oct 08 2011: (Start)
%F a(n) = 4*n - 3 + (-1)^n.
%F G.f.: 2*x^2*(3+x) / ( (1+x)*(x-1)^2 ). (End)
%F a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k)=2^(k+2) for k > 0. - _Philippe Deléham_, Oct 17 2011
%F a(n) = ceiling((8/3)*ceiling(3*n/2)). - _Clark Kimberling_, Jul 04 2012
%F Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - Pi/16. - _Amiram Eldar_, Dec 18 2021
%F E.g.f.: (4*x + 1)*exp(x) - exp(-x) = 4*x*exp(x) + 2*sinh(x). - _David Lovler_, Aug 02 2022
%t Array[8 # + {0, 6} &, 29, 0] // Flatten (* or *)
%t Rest@ CoefficientList[Series[2 x^2*(3 + x)/((1 + x) (x - 1)^2), {x, 0, 58}], x] (* _Michael De Vlieger_, Nov 18 2019 *)
%t LinearRecurrence[{1,1,-1},{0,6,8},80] (* _Harvey P. Dale_, Apr 09 2022 *)
%o (PARI) forstep(n=0,200,[6,2],print1(n", ")) \\ _Charles R Greathouse IV_, Oct 17 2011
%o (PARI) a(n) = 4*n - 3 + (-1)^n; \\ _David Lovler_, Jul 25 2022
%Y Union of A008590 and A017137.
%Y Cf. A030308, A047504 (complement).
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
|
