T.D. Noe 's nice one (A070897)

[T. D. Noe]

At 1:11 AM -0600 5/26/02, Don Reble wrote:
> > A070897 is really a nice & hard sequence.
>
>You might enjoy proving that the even-numbered terms are
>perfect squares. (Alas, that might dispel some of the
>mystery of the sequence.)
>
> > ... 1, 1, 1, 2, 4, 8, 36, 40, 49, 126, 121,
> > 440, 2809, 11395, 32761
>
>I concur. Here are the first 32 terms.
>
>1 1 1 1 2 4 8 36 40 49 126 121 440 2809 11395 32761 132183 881721
>3015500 19642624 106493895 249987721 1257922092 4609187881 29262161844
>189192811369 1068996265025 7388339422500 67416357342087 465724670229025
>1979950199225010 8532284145492496

Thanks for the extension.  I realized that for each n, the solution 
set can be decomposed into two sets: cases where the odd numbers are 
in 1..n or the odd numbers are in n+1..2n.  So there are actual two 
sequences whose product is A070897.  I submitted them yesterday:

%S A000001 1,1,1,1,2,2,4,6,10,7,21,11,40,53,215,181,773,939,3260,4432
%N A000001 Number of ways of pairing odd numbers in the range 1 to n 
with even numbers in the range n+1 to 2n such that each pair sums to 
a prime.
%C A000001 There is a similar sequence for even+odd numbers being 
submitted.  The product of the odd+even sequence and the even+odd 
sequence yields A070897.
%e A000001 a(5)=2 because there are two ways: 1+10,3+8,6+5 and 1+6,3+10,5+8
%O A000001 1


%S A000001 1,1,1,1,2,2,6,4,7,6,11,11,53,53,181,171,939,925,4432
%N A000001 Number of ways of pairing even numbers in the range 1 to n 
with odd numbers in the range n+1 to 2n such that each pair sums to a 
prime.
%C A000001 There is a similar sequence for odd+even numbers being 
submitted.  The product of the odd+even sequence and the even+odd 
sequence yields A070897.
%e A000001 a(6)=2 because there are two ways: 2+9,4+7,6+11 and 2+11,4+9,6+7
%O A000001 2


Note that the offset of the second sequence is 2.  As pointed out by 
Done Reble, it is easy to show that the even terms of these two 
sequences are the same, yielding squares in A070897.

Best regards,

Tony