The first few NSW primes are
7,
41,
239, 9369319, 63018038201, … (sequence A088165 in the
OEIS), corresponding to the indices 3, 5, 7, 19, 29, … (sequence A005850 in the
OEIS).
The sequence S alluded to in the formula can be described by the following
recurrence relation:
The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, … (sequence A001333 in the
OEIS). Each term in this sequence is half the corresponding term in the sequence of
companion Pell numbers. These numbers also appear in the
continued fraction convergents to
√2.