A053631 Pythagorean spiral: a(n-1)+1, a(n) and a(n)+1 are the sides of a right triangle (a primitive Pythagorean triangle).

2, 4, 12, 84, 3612, 6526884, 21300113901612, 226847426110843688722000884, 25729877366557343481074291996721923093306518970391612, 331013294649039928396936390888878360035026305412754995683702777533071737279144813617823976263475290370884

Offset: 1

Author

Henry Bottomley, Mar 21 2000

Keywords

• nonn

Comments

To derive a list of Pythagorean triples from this sequence, we note that the difference between the second and the third terms in the Pythagorean triple is 1 and that the last term of the previous triple gives us the first term in the next triple. Therefore the sequence is completely determined by the initial triple.

A053631 gives us a list of Pythagorean triples beginning with (3,4,5), since a(1)=2. Using any initial value h>1, (2h-1,2h^2-2h,2h^2-2h+1) forms a Pythagorean triple; we can use b(1)=2h-1 and the recursive formula b(n)=b(n-1)^2-b(n-1)+1 for n>1, we can create infinitely many of spirals of this type. - Haoqi Chen, Teena Carroll

Examples

			For n=3, a(n-1) = 4, so we want a right triangle with sides 4 + 1 = 5, a(n), and a(n)+1.  Solving (x+1)^2 = x^2 + 5^2 gives x = 12, so a(3) = 12. - _Michael B. Porter_, Jul 19 2016
		

Links

• Robert Israel, Table of n, a(n) for n = 1..13

Crossrefs

Apart from the initial term, the sequence is the same as A127690.

Cf. A046092

Programs

Maple

a[1]:= 2:
for n from 2 to 10 do a[n]:= a[n-1] + a[n-1]^2/2 od:
seq(a[i],i=1..10); # Robert Israel, Jul 08 2015

Mathematica

NestList[# + #^2/2 &, 2, 9] (* Robert G. Wilson v, Dec 12 2012 *)

Maxima

a[1]:2$
a[n]:=a[n-1] + (a[n-1]^2)/2$
A053631(n):=a[n]$
makelist(A053631(n),n,1,10); /* Martin Ettl, Nov 08 2012 */

PARI

main(size)={v=vector(size); v[1]=2;for(n=2,size,v[n]=v[n-1]+v[n-1]^2/2);return(v)} /* Anders Hellström, Jul 08 2015 */

Formula

a(1)=2; for n >= 2: a(n) = a(n-1) + a(n-1)^2/2 = A046092(a(n-1)/2).

a(n) = A053630(n) - 1. - Robert G. Wilson v, Jul 29 2014

a(n) = 2*A007018(n-1). - Ivan Neretin, Jul 26 2015

Extensions

Corrected and extended by James Sellers, Mar 22 2000

a(1) = 2 added by Zak Seidov, Apr 10 2007