A161482 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+151)^2 = y^2.

0, 96, 189, 453, 969, 1496, 3020, 6020, 9089, 17969, 35453, 53340, 105096, 207000, 311253, 612909, 1206849, 1814480, 3572660, 7034396, 10575929, 20823353, 40999829, 61641396, 121367760, 238964880, 359272749, 707383509, 1392789753

Offset: 1

Author

Klaus Brockhaus, Jun 13 2009

Keywords

• nonn

Comments

Corresponding values y of solutions (x, y) are in A161483.

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

lim_{n -> infinity} a(n)/a(n-1) = (187+78*sqrt(2))/151 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 0.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A161483, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151), A161485 (decimal expansion of (24723+6758*sqrt(2))/151^2).

Programs

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,96,189,453,969,1496,3020},30] (* Harvey P. Dale, Jul 10 2023 *)

PARI

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+302*n+22801), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+302 for n > 6; a(1)=0, a(2)=96, a(3)=189, a(4)=453, a(5)=969, a(6)=1496.

G.f.: x*(96+93*x+264*x^2-60*x^3-31*x^4-60*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 151*A001652(k) for k >= 0.