A160577 Positive numbers y such that y^2 is of the form x^2+(x+409)^2 with integer x.

305, 409, 641, 1189, 2045, 3541, 6829, 11861, 20605, 39785, 69121, 120089, 231881, 402865, 699929, 1351501, 2348069, 4079485, 7877125, 13685549, 23776981, 45911249, 79765225, 138582401, 267590369, 464905801, 807717425, 1559630965

Offset: 1

Author

Klaus Brockhaus, Jun 08 2009

Keywords

• nonn

Comments

(-136, a(1)) and (A129641(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 1.

Examples

			(-136, a(1)) = (-136, 305) is a solution: (-136)^2+(-136+409)^2 = 18496+74529 = 93025 = 305^2.
(A129641(1), a(2)) = (0, 409) is a solution: 0^2+(0+409)^2 = 167281 = 409^2.
(A129641(3), a(4)) = (611, 1189) is a solution: 611^2+(611+409)^2 = 373321+1040400 = 1413721 = 1189^2.
		

Links

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

Crossrefs

Cf. A129641, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{305,409,641,1189,2045,3541},50] (* or *) Select[Table[Sqrt[x^2+(x+409)^2],{x,-140,10^6}],IntegerQ] (* The second program generates the first 16 terms of the sequence. To generate more, increase the x constant but the program may take a long time to run. *) (* Harvey P. Dale, Mar 14 2022 *)

PARI

{forstep(n=-136, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=305, a(2)=409, a(3)=641, a(4)=1189, a(5)=2045, a(6)=3541.

G.f.: (1-x)*(305+714*x+1355*x^2+714*x^3+305*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 409*A001653(k) for k >= 1.