A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

0, 115, 136, 411, 1036, 1155, 2740, 6375, 7068, 16303, 37488, 41527, 95352, 218827, 242368, 556083, 1275748, 1412955, 3241420, 7435935, 8235636, 18892711, 43340136, 48001135, 110115120, 252605155, 279771448, 641798283, 1472291068, 1630627827, 3740674852

Offset

1,2

Comments

Also values x of Pythagorean triples (x, x+137, y).

Corresponding values y of solutions (x, y) are in A157213.

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

Limit_{n -> oo} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 0.

For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 25 a (prime) number, m >= 9 (p >= 137), the first three consecutive solutions are (0, p), (10*m+25, 2*m^2+10*m+25), (6*m^2-50*m+100, 10*m^2-70*m+125), the subsequent solutions are defined by the following recurrence relation: (X(n), Y(n)) = (3*X(n-3) + 2*Y(n-3) + p, 4*X(n-3) + 3*Y(n-3) + 2*p), X(n) = 6*X(n-3) - X(n-6) + 2*p, and Y(n) = 6*Y(n-3) - Y(n-6) (can be easily proved using X(n) = 3*X(n-3) + 2*Y(n-3) + p, and Y(n) = 4*X(n-3) + 3*Y(n-3) + 2*p). - Mohamed Bouhamida, Jun 18 2026

Links

Table of n, a(n) for n=1..31.

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)+274 for n > 6; a(1)=0, a(2)=115, a(3)=136, a(4)=411, a(5)=1036, a(6)=1155.

G.f.: x*(115+21*x+275*x^2-65*x^3-7*x^4-65*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 137*A001652(k) for k >= 0.

Example

For p=137 (m=9) the first three consecutive solutions are (0, 137), (115, 277), (136, 305).

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 115, 136, 411, 1036, 1155, 2740}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

Prog

(PARI) {forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ", ")))}

Crossrefs

Cf. A157213, A001652, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))), A129288, A129289, A129298.

Solutions x to x^2+(x+p)^2=y^2: A207058 (p=263), A129640 (p=313), this sequence (p=137).

Sequence in context: A036269 A036462 A020443 * A095621 A051975 A363790

Adjacent sequences: A129541 A129542 A129543 * A129545 A129546 A129547

Keyword

nonn,easy,changed

Author

Mohamed Bouhamida, May 30 2007

Extensions

Edited and extended by Klaus Brockhaus, Feb 25 2009

Status

approved