

A053630


Pythagorean spiral: a(n1), a(n)1 and a(n) are sides of a right triangle.


5




OFFSET

1,1


COMMENTS

a(n1)^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(n1)^2, so 2*a(n)1 = a(n1)^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(n1)^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(n1)^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(n1)^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



