%I #63 Sep 23 2023 12:35:11
%S 3,5,13,85,3613,6526885,21300113901613,226847426110843688722000885,
%T 25729877366557343481074291996721923093306518970391613
%N Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle.
%C 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
%D R. Gelca and T. Andreescu, Putnam and Beyond, Springer 2007, p. 121.
%F a(1) = 3, a(n) = (a(n-1)^2 + 1)/2 for n > 1.
%F 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).
%F a(1) = 3, a(n+1) = (1/4)*((a(n)-1)^2 + (a(n)+1)^2). - _Amarnath Murthy_, Aug 17 2005
%F 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
%F a(n) = (A006892(n+2) + 3)/2. - _Thomas Ordowski_, Jul 14 2014
%F a(n)^2 = A006892(n+3) + 2. - _Thomas Ordowski_, Jul 19 2014
%e a(3)=13 because 5,12,13 is a Pythagorean triple and a(2)=5.
%p A:= proc(n) option remember; (procname(n-1)^2+1)/2 end proc: A(1):= 3:
%p seq(A(n),n=1..10); # _Robert Israel_, Jul 14 2014
%t NestList[(#^2+1)/2&,3,10] (* _Harvey P. Dale_, Sep 15 2011 *)
%o (PARI) {a(n) = if( n>1, (a(n-1)^2 + 1) / 2, 3)} \\ _Michael Somos_, May 15 2011
%Y Cf. A000058, A001844, A006892.
%Y See also A018928, A180313 and A239381 for similar sequences with a(n) a leg and a(n+1) the hypotenuse of a Pythagorean triangle.
%Y Cf. A077125, A117191 (4^(1/Pi)).
%K nonn,easy
%O 1,1
%A _Henry Bottomley_, Mar 21 2000
%E Corrected and extended by _James A. Sellers_, Mar 22 2000