A115599 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z-X values.

2, 9, 50, 289, 1682, 9801, 57122, 332929, 1940450, 11309769, 65918162, 384199201, 2239277042, 13051463049, 76069501250, 443365544449, 2584123765442, 15061377048201, 87784138523762, 511643454094369, 2982076586042450, 17380816062160329, 101302819786919522

Offset: 1

Author

N. J. A. Sloane, Mar 14 2006

Keywords

• nonn
• easy

Comments

Old name was A001653(n) - A001652(n).

Links

• L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213.
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Identical to A055997 without leading term.

Programs

Mathematica

CoefficientList[Series[(x^2 - 5 x + 2)/((1 - x) (x^2 - 6 x + 1)), {x, 0, 33}], x] (* Vincenzo Librandi, Nov 26 2015 *)
LinearRecurrence[{7,-7,1},{2,9,50},30] (* Harvey P. Dale, Jan 17 2017 *)

PARI

Vec(-x*(x^2-5*x+2)/((x-1)*(x^2-6*x+1)) + O(x^100)) \\ Altug Alkan, Nov 26 2015

Formula

a(n) = (2+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/4.

a(n) = 7*a(n-1)-7*a(n-2)+a(n-3).

G.f.: -x*(x^2-5*x+2) / ((x-1)*(x^2-6*x+1)).

Extensions

Corrected and edited by Colin Barker, Jul 31 2013