This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053630 #73 Jul 13 2025 19:34:12
%S A053630 3,5,13,85,3613,6526885,21300113901613,226847426110843688722000885,
%T A053630 25729877366557343481074291996721923093306518970391613
%N A053630 Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle.
%C A053630 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 A053630 R. Gelca and T. Andreescu, Putnam and Beyond, Springer 2007, p. 121.
%H A053630 Steven Finch, <a href="https://arxiv.org/abs/2411.16062">Exercises in Iterational Asymptotics</a>, arXiv:2411.16062 [math.NT], 2024. See p. 10.
%H A053630 Miguel-Ángel Pérez García-Ortega, <a href="/A365577/a365577.pdf">Capitulo 5. Catetos</a>, El Libro de las Ternas Pitagóricas.
%F A053630 a(1) = 3, a(n) = (a(n-1)^2 + 1)/2 for n > 1.
%F A053630 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 A053630 a(1) = 3, a(n+1) = (1/4)*((a(n)-1)^2 + (a(n)+1)^2). - _Amarnath Murthy_, Aug 17 2005
%F A053630 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 A053630 a(n) = (A006892(n+2) + 3)/2. - _Thomas Ordowski_, Jul 14 2014
%F A053630 a(n)^2 = A006892(n+3) + 2. - _Thomas Ordowski_, Jul 19 2014
%e A053630 a(3)=13 because 5,12,13 is a Pythagorean triple and a(2)=5.
%p A053630 A:= proc(n) option remember; (procname(n-1)^2+1)/2 end proc: A(1):= 3:
%p A053630 seq(A(n),n=1..10); # _Robert Israel_, Jul 14 2014
%t A053630 NestList[(#^2+1)/2&,3,10] (* _Harvey P. Dale_, Sep 15 2011 *)
%o A053630 (PARI) {a(n) = if( n>1, (a(n-1)^2 + 1) / 2, 3)}; /* _Michael Somos_, May 15 2011 */
%Y A053630 Cf. A000058, A001844, A006892.
%Y A053630 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 A053630 Cf. A077125, A117191 (4^(1/Pi)).
%K A053630 nonn,easy
%O A053630 1,1
%A A053630 _Henry Bottomley_, Mar 21 2000
%E A053630 Corrected and extended by _James Sellers_, Mar 22 2000