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A115243
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G.f.: (4*x^2 + 2*x)/(4*x^3 - x^2 - 4*x + 1).
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2
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0, 2, 12, 50, 204, 818, 3276, 13106, 52428, 209714, 838860, 3355442, 13421772, 53687090, 214748364, 858993458, 3435973836, 13743895346, 54975581388, 219902325554, 879609302220, 3518437208882, 14073748835532, 56294995342130, 225179981368524, 900719925474098
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OFFSET
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0,2
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COMMENTS
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Inverse Z-transform of polynomial in A112627.
a(n) is also the number of corners in the n-th approximation of the Hilbert Curve. The 1st Hilbert Curve approximation has 2 corners. To find a(n) given a(n - 1), look at how the n-th Hilbert Curve approximation is constructed: duplicate the (n-1)-th approximation 4 times and connect the duplicates with 3 line segments. a(n) will always be 4 * a(n - 1) corners from the 4 duplicates plus 4 new corners if n is even or 2 new corners if n is odd. - Mikel Mcdaniel, Jan 10 2019
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LINKS
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FORMULA
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a(n) = InverseZTransform[(1 + 2*x)/(1 - x - 16*x^2 + 16*x^3), x, n] * 2^(2*n).
a(n) = 5*a(n-1)-4*a(n-2) +2*(-1)^n.
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MAPLE
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seq((4^(n+1)+(-1)^n)/5 - 1, n=0..100); # Robert Israel, Mar 09 2016
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MATHEMATICA
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Table[InverseZTransform[(1 + 2*x)/(1 - x - 16*x^2 + 16*x^3), x, n]*2^( 2*n), {n, 1, 25}]
LinearRecurrence[{4, 1, -4}, {0, 2, 12}, 50] (* G. C. Greubel, Feb 07 2016 *)
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PROG
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(PARI) a(n) = (bitneg(0, 2*n+2)-1)\5; \\ Kevin Ryde, May 05 2023
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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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EXTENSIONS
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STATUS
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approved
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