# [seqfan] Interesting sequence

John W. Layman layman at math.vt.edu
Tue Jan 19 18:32:35 CET 2010

```I came up with the following sequence recently and found it rather
interesting.

Number of subsets S of {1,2,3,...,n} with the property that if x is a
member of S then at least one of x-2 and x+2 is also a member of S:
{a(n)}={1, 1, 2, 4, 8, 16, 28, 49, 84, 144, 252, 441, 777, 1369, 2405,
4225, 7410, 12996, 22800, 40000, 70200, 123201, 216216, 379456, 665896,
1168561, 2050657, 3598609, 6315113, 11082241, 19448018, 34128964,...},
for n=1,2,3,...

Note the even-index bisection:  {1, 4, 16, 49, 144, 441, 1369, ...},
which gives the SQUARES of:
A005251 = {0, 1, 1, 1, 2, 4, 7, 12, 21, 37, ...}, n=0,1,2,..., beginning
with A005251(3).

The second difference of {a(n)} is
{d2(n)} = {1, 1, 2, 4, 4, 9, 14, 25, 48, 81, 147, 256, 444, 784, 1365,
2401, 4218, 7396, 13000, 22801, 40014, 70225, 123200, 216225, 379431,
665856, 1168552, 2050624, 3598649, 6315169,...}, which also has the
property that the even-index terms are SQUARES:  {1, 4, 9, 25, 81, 256,
784, ...}.  In fact, they are the squares of
A005314 = {0, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151, ...}, n=0, 1, 2, ...,
beginning with A005314(1).

Also, it appears that A005314 is the first difference of A005251,
beginning with A005251(2)..

I will submit this once the current submission hiatus is over.

It would be nice to have proofs of the "squares" observations pointed
out above.  Also formula or recurrence, etc. of {a(n)}.

Best regards,
John

```