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

[Ed Jeffery]

Doug,

Your first few terms, when factored and written in the form

A(n)*(B(n) + C(n)*sqrt(2)),

are

 n
 1 |              4*(        1 +         1*sqrt(2))
 2 |             48*(        3 +         2*sqrt(2))
 3 |            576*(       17 +        12*sqrt(2))
 4 |          16128*(       79 +        56*sqrt(2))
 5 |         193536*(     1381 +       976*sqrt(2))
 6 |       25546752*(     3213 +      2272*sqrt(2))
 7 |     2145927168*(    16201 +     11456*sqrt(2))
 8 |     8583708672*(  2262209 +   1599616*sqrt(2))
 9 |   103004504064*(134247851 +  94927616*sqrt(2))
10 | 23485026926592*(520704987 + 368194048*sqrt(2)).

I was hoping you might find a recurrence relation (or some other clue) in
one of those sequences, since it appears that

B(n)/C(n) -> sqrt(2)

with increasing n. These three sequences are not in OEIS either; and of
course there is the possibility of {A(n)/4}, if you can somehow prove it.

LEJ