A159701 Positive numbers y such that y^2 is of the form x^2+(x+967)^2 with integer x.

925, 967, 1013, 4537, 4835, 5153, 26297, 28043, 29905, 153245, 163423, 174277, 893173, 952495, 1015757, 5205793, 5551547, 5920265, 30341585, 32356787, 34505833, 176843717, 188589175, 201114733, 1030720717, 1099178263

Offset

1,1

Comments

(-43, a(1)) and (A130017(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.

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

Lim_{n -> infinity} a(n)/a(n-1) = (969+44*sqrt(2))/967 for n mod 3 = {0, 2}.

Lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 1.

Links

G. C. Greubel, Table of n, a(n) for n = 1..3895

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)=925, a(2)=967, a(3)=1013, a(4)=4537, a(5)=4835, a(6)=5153.

G.f.: (1-x)*(925+1892*x+2905*x^2+1892*x^3+925*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 967*A001653(k) for k >= 1.

Example

(-43, a(1)) = (-43, 925) is a solution: (-43)^2+(-43+967)^2 = 1849+853776 = 855625 = 925^2.
(A130017(1), a(2)) = (0, 967) is a solution: 0^2+(0+967)^2 = 935089 = 967^2.
(A130017(3), a(4)) = (2688, 4537) is a solution: 2688^2+(2688+967)^2 = 7225344+13359025 = 20584369 = 4537^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {925, 967, 1013, 4537, 4835, 5153}, 40] (* G. C. Greubel, May 22 2018 *)

Prog

(PARI) {forstep(n=-44, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089, &k), print1(k, ", ")))};
(PARI) x='x+O('x^30); Vec((1-x)*(925+1892*x+2905*x^2+1892*x^3+925*x^4)/( 1-6*x^3+x^6)) \\ G. C. Greubel, May 22 2018
(Magma) I:=[925, 967, 1013, 4537, 4835, 5153]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018

Crossrefs

Cf. A130017, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44*sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

Sequence in context: A172880 A172912 A172702 * A115696 A066741 A325141

Adjacent sequences: A159698 A159699 A159700 * A159702 A159703 A159704

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 21 2009

Status

approved