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

Offset

1,2

Comments

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

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

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

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

lim_{n -> infinity} 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).

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*(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) [From Harvey P. Dale, May 13 2012]

Mathematica

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

Prog

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

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).

Sequence in context: A293512 A373191 A275237 * A304837 A231089 A237318

Adjacent sequences: A129639 A129640 A129641 * A129643 A129644 A129645

Keyword

nonn,easy

Author

Mohamed Bouhamida, May 31 2007

Extensions

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

Status

approved