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A356880
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Squares that can be expressed as the sum of two powers of two (2^x + 2^y).
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = 9 * 2^(n-2) if n is even (see A002063).
a(n) = 2^(n+1) if n is odd (see A000302).
G.f.: x*(4 + 9*x)/(1 - 4*x^2).
E.g.f.: (9*(cosh(2*x) - 1) + 8*sinh(2*x))/4. (End)
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EXAMPLE
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2^4 + 2^7 = 144, a square, thus 144 is a term.
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MAPLE
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seq(`if`(n::even, 9*2^(n-2), 2^(n+1)), n=1..50); # Robert Israel, Sep 15 2022
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MATHEMATICA
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Select[Range[2, 2^17]^2, DigitCount[#, 2, 1] <= 2 &] (* Amiram Eldar, Sep 03 2022 *)
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PROG
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(Python)
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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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