%I #38 Jul 05 2018 14:00:33

%S 8,0,9,0,8,0,4,0,0,0,16,0,128,0,576,0,2048,0,6400,0,18432,0,50176,0,

%T 131072,0,331776,0,819200,0,1982464,0,4718592,0,11075584,0,25690112,0,

%U 58982400,0,134217728,0,303038464,0,679477248,0

%N a(n+6) = 6*a(n+4) - 12*a(n+2) + 8*a(n), a(0)..a(5) = 8,0,9,0,8,0.

%C The infinite sum of the reciprocals of the even terms a(2*k) for k>=5 yields (1/96)*(Pi^2 - 6*log^2(2)). That is, sum(k>=1, 1/(k^2*2^k) = dilog(1/2) = (Pi^2 - 6*(log(2))^2)/12. See the Jolley reference, pp. 66-69, (360) (c), and Abramowitz-Stegun, p. 1004, 27.7.3 for x=1/2. For the decimal expansion of dilog(1/2) see A076788.

%D L. B. W. Jolley, Summation of Series, Dover (1961).

%H Vincenzo Librandi, <a href="/A243456/b243456.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H G. Myerson and A. J. van der Poorten, <a href="https://www.jstor.org/stable/2974639">Some problems concerning recurrence sequences</a>, Amer. Math. Monthly 102 (1995), no. 8, 698-705.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-12,0,8).

%F a(n) = 0, if n is odd, a(n) = (n - 8)^2*2^((n-6)/2), if n is even.

%F a(n) = 2^(n/2 - 4)*(1 + (-1)^n)*(n - 8)^2.

%F a(n+2) = (2*(n-6)^2*a(n))/(n-8)^2, n>=0, with a(0) = 8, a(1) = 0 and a(10) = 0/0 := 16.

%F G.f.: (8 - 39*x^2 + 50*x^4)/(1-2*x^2)^3.

%p A243456:=n->2^(n/2 - 4)*(1 + (-1)^n)*(n - 8)^2; seq(A243456(n), n=0..50); # _Wesley Ivan Hurt_, Jun 08 2014

%t Table[2^(n/2 - 4)*(1 + (-1)^n)*(n - 8)^2, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jun 08 2014 *)

%t CoefficientList[Series[(8 - 39 x^2 + 50 x^4)/(1 - 2 x^2)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 16 2014 *)

%t LinearRecurrence[{0,6,0,-12,0,8},{8,0,9,0,8,0},50] (* _Harvey P. Dale_, Mar 02 2016 *)

%Y Cf. A076788.

%K nonn

%O 0,1

%A _Alexander R. Povolotsky_, Jun 05 2014

%E Comment on the sum reformulated and Jolley and Abramowitz-Stegun reference added. G.f. corrected for offset 0. In the recurrence a(10) defined. - _Wolfdieter Lang_, Jun 16 2014