A130017 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.

0, 45, 2688, 2901, 3128, 18105, 19340, 20657, 107876, 115073, 122748, 631085, 673032, 717765, 3680568, 3925053, 4185776, 21454257, 22879220, 24398825, 125046908, 133352201, 142209108, 728829125, 777235920, 828857757, 4247929776

Offset: 1

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+967, y).

Corresponding values y of solutions (x, y) are in A159701.

For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.

lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

lim_{n -> infinity} a(n)/a(n-1) = (969+44**sqrt(2))/967 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 0.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A159701, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44**sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,45,2688,2901,3128,18105,19340},40] (* Harvey P. Dale, Nov 03 2013 *)

PARI

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1934 for n > 6; a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105.

G.f.: x*(45+2643*x+213*x^2-43*x^3-881*x^4-43*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 967*A001652(k) for k >= 0.

a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105, a(7)=19340, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Nov 03 2013

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 21 2009