[seqfan] Sum of Pell numbers and companion Pell numbers

[Robert Dougherty-Bliss]

Dear Sequence Fans,

Consider the sequence a(n) which satisfies the recurrence
     a(n) = 2 * a(n - 1) + a(n - 2)
with initial conditions a(0) = a(1) = 2. (This is A2203, the companion
Pell numbers.)

The sequence of positive n such that a(n) = 2 (mod n) begins
1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 29.
Except for the leading 1 and 2, this looks like A270342, the sequence
of n such that n divides the sum of the first n Pell numbers, but
there is no comment to this effect. Can anyone prove this?

The sum of the first n Pell numbers S(n) turns out to equal (a(n) - 2)
/ 4, so everything in A270342 is in this sequence. For the other way,
I have managed to prove that if a(n) = 2 (mod n) and n is odd or
divisible by 4, then n is in A270341, but I cannot figure out the
"divisible by just 2" case.

Robert