A053630 Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle.

3, 5, 13, 85, 3613, 6526885, 21300113901613, 226847426110843688722000885, 25729877366557343481074291996721923093306518970391613

Offset: 1

Author

Henry Bottomley, Mar 21 2000

Keywords

• nonn
• easy

Comments

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

Examples

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

References

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

Links

• Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 10.
• Miguel-Ángel Pérez García-Ortega, Capitulo 5. Catetos, El Libro de las Ternas Pitagóricas.

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.

Cf. A077125, A117191 (4^(1/Pi)).

Programs

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 *)

PARI

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

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

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

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

Extensions

Corrected and extended by James Sellers, Mar 22 2000