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A053630 Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right angled triangle. 5
3, 5, 13, 85, 3613, 6526885, 21300113901613, 226847426110843688722000885, 25729877366557343481074291996721923093306518970391613 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n-1)^2 + (a(n)-1)^2 = a(n)^2 with a(1) = 3.

Least prime factors of a(n):

3, 5, 13, 5, 3613, 5, 233, 5, 3169, 5, 101, 5, 29, 5, 695838629, 5, 1217, 5, 2557, 5, 101, 5, 769, 5. - Zak Seidov, Nov 11 2013

We have a(n)^2-(a(n)-1)^2 = a(n-1)^2, so 2*a(n)-1 = a(n-1)^2, and see the first formula. - Thomas Ordowski, Jul 13 2014

REFERENCES

R. Gelca and T. Andreescu, Putnam and Beyond, Springer 2007, p. 121

LINKS

Table of n, a(n) for n=1..9.

FORMULA

a(1) = 3, a(n) = (a(n-1)^2 + 1)/2 for n > 1.

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, Aug 17 2005

a(n) = (A006892(n+2)+3)/2. - Thomas Ordowski, Jul 14 2014

a(n)^2 = A006892(n+3) + 2. - Thomas Ordowski, Jul 19 2014

EXAMPLE

a(3)=13 because 5,12,13 is a Pythagorean triple and a(2)=5.

MAPLE

A:= proc(n) option remember; (procname(n-1)^2+1)/2 end proc: A(1):= 3:

seq(A(n), n=1..10); # Robert Israel, Jul 14 2014

MATHEMATICA

NestList[(#^2+1)/2&, 3, 10] (* Harvey P. Dale, Sep 15 2011 *)

PROG

(PARI) {a(n) = if( n>1, (a(n-1)^2 + 1) / 2, 3)} \\ Michael Somos, May 15 2011

CROSSREFS

Cf. A000058, A001844, A006892.

See also A018928, A180313 and A239381 for similar sequences with a(n) a leg and a(n+1) the hypotenuse of a Pythagorean triangle.

Sequence in context: A018928 A239381 A180313 * A155012 A121533 A187733

Adjacent sequences:  A053627 A053628 A053629 * A053631 A053632 A053633

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Mar 21 2000

EXTENSIONS

Corrected and extended by James A. Sellers, Mar 22 2000

STATUS

approved

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Last modified December 5 15:11 EST 2019. Contains 329753 sequences. (Running on oeis4.)