A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

29, 41, 85, 89, 205, 481, 505, 1189, 2801, 2941, 6929, 16325, 17141, 40385, 95149, 99905, 235381, 554569, 582289, 1371901, 3232265, 3393829, 7996025, 18839021, 19780685, 46604249, 109801861, 115290281, 271629469, 639972145, 671961001

Offset: 1

Author

Klaus Brockhaus, Feb 26 2009

Keywords

• nonn
• easy

Comments

(-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2 for n mod 3 = 1.

Examples

			(-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.
(A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.
(A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1001
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A129288, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // G. C. Greubel, Feb 04 2018

Mathematica

CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x,0,50}], x] (* G. C. Greubel, Feb 04 2018 *)

PARI

{forstep(n=-20, 500000000, [3 ,1], if(issquare(n^2+(n+41)^2, &k), print1(k, ",")))}

PARI

x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ G. C. Greubel, Feb 04 2018

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.

G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 41*A001653(k) for k >= 1.