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A053630
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Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right angled triangle.
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1
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OFFSET
| 1,1
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COMMENTS
| a(3)=85 because 13,84,85 is a Pythagorean triple and a(2)=13
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REFERENCES
| R. Gelca and T. Andreescu, Putnam and Beyond, Springer 2007, p. 121
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FORMULA
| a(n) = (a(n-1)^2 + 1) / 2.
a(n) = 2*A000058(n)-1 =A053631(n)+1 =floor[2 * 1.597910218031873...^(2^n)]. Constructing the spiral as a sequence of triangles with one vertex at the origin, then for large n the other vertices are close to lying on the doubly logarithmic spiral r=2*2.228918357655...^(1.5546822754821...^theta) where theta(n)=n*pi/2-1.215918200344... and 1.5546822754821...=4^(1/pi).
a(1) = 3, a(n+1) = (1/4)[{a(n)-1}^2 + {a(n)+1}^2] - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 17 2005
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MATHEMATICA
| NestList[(#^2+1)/2&, 3, 10] (* From Harvey P. Dale, Sep 15 2011 *)
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PROG
| (PARI) {a(n) = if( n<=1, 3 * (n==1), (a(n-1)^2 + 1) / 2)} /* Michael Somos May 15 2011 */
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CROSSREFS
| Cf. A000058, A001844.
Sequence in context: A051901 A018928 A180313 * A155012 A121533 A087170
Adjacent sequences: A053627 A053628 A053629 * A053631 A053632 A053633
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KEYWORD
| nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Mar 21 2000
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EXTENSIONS
| Corrected and extended by James A. Sellers (sellersj(AT)math.psu.edu), Mar 22 2000.
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