A239381 -id:A239381 - cp's OEIS frontend

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

Henry Bottomley, Mar 21 2000

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

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

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

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.

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)).

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

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

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

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

Corrected and extended by James Sellers, Mar 22 2000

A307077 Let a(1)=3; for n > 1, let a(n) be the least positive integer k such that k > a(n-1), a(1)^2 + ... + a(n-1)^2 + k^2 is a square and the Pythagorean triple sqrt(a(1)^2 + ... + a(n-1)^2), a(n), sqrt(a(1)^2 + ... + a(n)^2) is primitive.

Original entry on oeis.org

3, 4, 12, 84, 132, 12324, 89892, 2447844, 28350372, 295742791596, 171480834409712412, 633511848768467916, 1616599508725767821225590810932, 4158520496012961741299012805876, 115366949386695884000892071516523067413910188

Offset: 1

Rohan Hemasingha, May 30 2019

nonn

For n > 1, a(n) is the even value of a primitive Pythagorean triple where the larger odd value of the triple equals the smaller odd value of a primitive Pythagorean triple with even value a(n+1) (see A239381). - Torlach Rush, Jan 27 2023

Cf. A018930, A127689, A239381.

lista(NN) = s=9;k=3;print1(k);for(n=1,NN-1,v=divisors(s);j=#v;while(v[j]*(v[j]+2*k)>s,j--);while(gcd((s-v[j]^2)/(2*v[j]), s)!=1, j--);print1(", ", k=(s-v[j]^2)/(2*v[j]));s+=k^2); \\ Jinyuan Wang, May 31 2019

The numbers are generated by using the well-known characterization of primitive Pythagorean triples, namely (a,b,c) is a PPT iff there are positive integers j,k of opposite parity with j > k, and gcd(j,k)=1 such that a = j^2 - k^2, b = 2jk, c = j^2 + k^2.

a(14)-a(15) from Jinyuan Wang, Jun 01 2019

cp's OEIS Frontend

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

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