[seqfan] Re: Some Remarkable Series for A193543(n), A193544(n)

[Paul D Hanna]

SeqFans, 
    Here are a few more terms; given  
 a(n) = (sqrt(2)*Pi/L) * Sum_{k>=1} (2*k*Pi/L)^(2*n) / cosh(k*Pi/2) for n>0 with a(0)=1 
 
where L = Lemniscate constant = 2*(Pi/2)^(3/2) / gamma(3/4)^2 
 
then  
a(1) = 4 + 4*sqrt(2)) 
a(2) = 144 + 96*sqrt(2) 
a(3) = 9792 + 6912*sqrt(2) 
a(4) = 1274112 + 903168*sqrt(2)
a(5) = 267273216 + 188891136*sqrt(2)
a(6) = 82081714176 + 58042220544*sqrt(2)
... 
which supports the conjecture that 
  a(n) = b(n) + c(n)*sqrt(2)
where b(n) and c(n) are integeral. 
 
Can someone guess a formula or generate more terms for b(n) and c(n)? 
 
Thanks, 
    Paul 

---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Some Remarkable Series for A193543(n), A193544(n)
Date: Fri, 31 Aug 2012 02:32:08 GMT

SeqFans,      
    Consider a slight variation in the series for A193543(n) mentioned in prior email 
by defining  
a(n) = (sqrt(2)*Pi/L) * Sum_{k>=1} (2*k*Pi/L)^(2*n) / cosh(k*Pi/2) for n>0 with a(0)=1. 
Then 
a(1) = 9.656854249492380195206754896838792314278687501507792...
a(2) = 279.7645019878171246849621175241310155426885000361870...
a(3) = 19567.04414312283297731727246173743311907357200260546...
a(4) = 2551384.434701383509036123601667024594225613408340447...
a(5) = 534405622.3432607796041264218343605722780425756872021...
a(6) = 164165809659.5750828344220598502172879799939350603429...
a(7) = 69532826885135.18156333436878613710259344193824133284...
a(8) = 38836197110487315.84344087627800470932336285596433235...
a(9) = 27656273940212637887.65845519441121088691142638967441...
a(10) = 24457541968339546203692.3434925846139712563531814594...
... 
which I find to begin:   
a(1) = 4 + 4*sqrt(2)) 
a(2) = 144 + 96*sqrt(2) 
a(3) = 9792 + 6912*sqrt(2) 
... 
>From this I conjecture that 
a(n) = b(n) + c(n)*sqrt(2) 

Without Plouffe's converter, I can't go much further. 
(BTW, it was by using Plouffe's Inverter that I was able to find  
the constants involved in the sums for A193543(n), A193544(n).)

Can someone find more terms of b(n) and c(n)? 

Here is my PARI code:\\p80 
{a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2); if(n==0,1, 
sqrt(2)*Pi/L*suminf(k=1, (2*k*Pi/L)^(2*n)/cosh(k*Pi/2)) )} 
for(n=0, 20, print("a("n") = "a(n))) 

Thanks, 
    Paul