A160209 Positive numbers y such that y^2 is of the form x^2+(x+937)^2 with integer x.

673, 937, 1685, 2353, 4685, 9437, 13445, 27173, 54937, 78317, 158353, 320185, 456457, 922945, 1866173, 2660425, 5379317, 10876853, 15506093, 31352957, 63394945, 90376133, 182738425, 369492817, 526750705, 1065077593, 2153561957

Offset

1,1

Comments

(-385, a(1)) and (A129974(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 1.

Links

Table of n, a(n) for n=1..27.

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

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=673, a(2)=937, a(3)=1685, a(4)=2353, a(5)=4685, a(6)=9437.

G.f.: (1-x)*(673+1610*x+3295*x^2+1610*x^3+673*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 937*A001653(k) for k >= 1.

Example

(-385, a(1)) = (-385, 673) is a solution: (-385)^2+(-385+937)^2 = 148225+304704 = 452929 = 673^2.
(A129974(1), a(2)) = (0, 937) is a solution: 0^2+(0+937)^2 = 877969 = 937^2.
(A129974(3), a(4)) = (1128, 2353) is a solution: 1128^2+(1128+937)^2 = 1272384+4264225 = 5536609 = 2353^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {673, 937, 1685, 2353, 4685, 9437}, 30] (* Harvey P. Dale, Dec 25 2017 *)

Prog

(PARI) {forstep(n=-388, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969, &k), print1(k, ", ")))}

Crossrefs

Cf. A129974, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).

Sequence in context: A047728 A297123 A335254 * A234117 A171266 A267818

Adjacent sequences: A160206 A160207 A160208 * A160210 A160211 A160212

Keyword

nonn

Author

Klaus Brockhaus, May 18 2009

Status

approved