A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

0, 348, 495, 1371, 3255, 4088, 9140, 20096, 24947, 54383, 118235, 146508, 318072, 690228, 855015, 1854963, 4024047, 4984496, 10812620, 23454968, 29052875, 63021671, 136706675, 169333668, 367318320, 796785996, 986950047, 2140889163, 4644010215, 5752367528

Offset: 1

Author

Mohamed Bouhamida, May 31 2007

Keywords

• nonn
• easy
• changed

Comments

Also values x of Pythagorean triples (x, x+457, y).

Corresponding values y of solutions (x, y) are in A160580.

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

Limit_{n->oo} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {1, 2}.

Limit_{n->oo} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 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. A160580, A001652, A129641, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,348,495,1371,3255,4088,9140},30] (* Harvey P. Dale, May 13 2012 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+914 for n > 6; a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088.

G.f.: x^2*(348+147*x+876*x^2-204*x^3-49*x^4-204*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 457*A001652(k) for k >= 0.

a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088, a(7)=9140, a(n) = a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, May 13 2012

Extensions

Edited and two terms added by Klaus Brockhaus, Jun 08 2009