OFFSET
1,1
COMMENTS
If x is even, y = x + 3; if x is odd, y = x.
Proof for odd x: (2^odd + 2^odd) = 2^(odd + 1) = 2^even --> must be a square.
Proof for even x: 2^even + 2^(even + 3) = 1*(2^even) + (2^even * 2^3) = 1*(2^even) + (2^even * 8) = 1*(2^even) + 8*(2^even) = 9*(2^even); since 9 is a square and 2^even is a square, the multiplication result must be a square too.
And 9 is the only square that can be written as 1 + a power of 2.
Note that a(n) = A272711(n+1) for n=1..23, but beyond it differs more and more.
LINKS
FORMULA
a(n) = A029744(n+1)^2.
a(n) = 9 * 2^(n-2) if n is even (see A002063).
a(n) = 2^(n+1) if n is odd (see A000302).
From Stefano Spezia, Sep 09 2022: (Start)
G.f.: x*(4 + 9*x)/(1 - 4*x^2).
E.g.f.: (9*(cosh(2*x) - 1) + 8*sinh(2*x))/4. (End)
EXAMPLE
2^4 + 2^7 = 144, a square, thus 144 is a term.
MAPLE
seq(`if`(n::even, 9*2^(n-2), 2^(n+1)), n=1..50); # Robert Israel, Sep 15 2022
MATHEMATICA
Select[Range[2, 2^17]^2, DigitCount[#, 2, 1] <= 2 &] (* Amiram Eldar, Sep 03 2022 *)
PROG
(Python)
def A356880(n):
if n % 2 == 0: return 9*2**(n-2)
else: return 2**(n+1)
(PARI) a(n) = if (n%2, 2^(n+1), 9*2^(n-2)); \\ Michel Marcus, Sep 15 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karl-Heinz Hofmann, Sep 02 2022
STATUS
approved