A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

4, 9, 16, 36, 64, 144, 256, 576, 1024, 2304, 4096, 9216, 16384, 36864, 65536, 147456, 262144, 589824, 1048576, 2359296, 4194304, 9437184, 16777216, 37748736, 67108864, 150994944, 268435456, 603979776, 1073741824, 2415919104, 4294967296, 9663676416, 17179869184

Offset: 1

Author

Karl-Heinz Hofmann, Sep 02 2022

Keywords

• nonn
• easy

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.

Examples

			2^4 + 2^7 = 144, a square, thus 144 is a term.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,4).

Crossrefs

Cf. A029744, A000302, A002063.

Intersection of A000290 and A048645\{1}.

Cf. A272711, A270473 (squares that can be expressed as 3^x + 3^y).

Cf. A220221.

Programs

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 *)

PARI

a(n) = if (n%2, 2^(n+1), 9*2^(n-2)); \\ Michel Marcus, Sep 15 2022

Python

def A356880(n):
    if n % 2 == 0: return 9*2**(n-2)
    else: return 2**(n+1)

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)