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

%I A356880 #50 Sep 25 2022 09:34:29
%S A356880 4,9,16,36,64,144,256,576,1024,2304,4096,9216,16384,36864,65536,
%T A356880 147456,262144,589824,1048576,2359296,4194304,9437184,16777216,
%U A356880 37748736,67108864,150994944,268435456,603979776,1073741824,2415919104,4294967296,9663676416,17179869184
%N A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).
%C A356880 If x is even, y = x + 3; if x is odd, y = x.
%C A356880 Proof for odd x: (2^odd + 2^odd) = 2^(odd + 1) = 2^even --> must be a square.
%C A356880 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.
%C A356880 And 9 is the only square that can be written as 1 + a power of 2.
%C A356880 Note that a(n) = A272711(n+1) for n=1..23, but beyond it differs more and more.
%H A356880 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4).
%F A356880 a(n) = A029744(n+1)^2.
%F A356880 a(n) = 9 * 2^(n-2) if n is even (see A002063).
%F A356880 a(n) = 2^(n+1) if n is odd (see A000302).
%F A356880 From _Stefano Spezia_, Sep 09 2022: (Start)
%F A356880 G.f.: x*(4 + 9*x)/(1 - 4*x^2).
%F A356880 E.g.f.: (9*(cosh(2*x) - 1) + 8*sinh(2*x))/4. (End)
%e A356880 2^4 + 2^7 = 144, a square, thus 144 is a term.
%p A356880 seq(`if`(n::even, 9*2^(n-2), 2^(n+1)),n=1..50); # _Robert Israel_, Sep 15 2022
%t A356880 Select[Range[2, 2^17]^2, DigitCount[#, 2, 1] <= 2 &] (* _Amiram Eldar_, Sep 03 2022 *)
%o A356880 (Python)
%o A356880 def A356880(n):
%o A356880     if n % 2 == 0: return 9*2**(n-2)
%o A356880     else: return 2**(n+1)
%o A356880 (PARI) a(n) = if (n%2, 2^(n+1), 9*2^(n-2)); \\ _Michel Marcus_, Sep 15 2022
%Y A356880 Cf. A029744, A000302, A002063.
%Y A356880 Intersection of A000290 and A048645\{1}.
%Y A356880 Cf. A272711, A270473 (squares that can be expressed as 3^x + 3^y).
%Y A356880 Cf. A220221.
%K A356880 nonn,easy
%O A356880 1,1
%A A356880 _Karl-Heinz Hofmann_, Sep 02 2022