A159690 Positive numbers y such that y^2 is of the form x^2+(x+881)^2 with integer x.

841, 881, 925, 4121, 4405, 4709, 23885, 25549, 27329, 139189, 148889, 159265, 811249, 867785, 928261, 4728305, 5057821, 5410301, 27558581, 29479141, 31533545, 160623181, 171817025, 183790969, 936180505, 1001423009, 1071212269

Offset: 1

Author

Klaus Brockhaus, Apr 21 2009

Keywords

• nonn
• easy

Comments

(-41,a(1)) and (A130014(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (883+42*sqrt(2))/881 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 1.

Examples

			(-41, a(1)) = (-41, 841) is a solution: (-41)^2+(-41+881)^2 = 1681+705600 = 707281 = 841^2.
(A130014(1), a(2)) = (0, 881) is a solution: 0^2+(0+881)^2 = 776161 = 881^2.
(A130014(3), a(4)) = (2440, 4121) is a solution: 2440^2+(2440+881)^2 = 5953600+11029041 = 16982641 = 4121^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..3501
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A130014, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

Programs

Magma

I:=[841, 881, 925, 4121, 4405, 4709]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 02 2018

Mathematica

CoefficientList[Series[(1 - x)*(841 + 1722*x + 2647*x^2 + 1722*x^3 + 841*x^4)/(1 - 6*x^3 + x^6), {x,0,50}], x] (* or *) LinearRecurrence[{0, 0,6,0,0,-1}, {841, 881, 925, 4121, 4405, 4709}, 30] (* G. C. Greubel, Jun 02 2018 *)

PARI

{forstep(n=-44, 10000000, [3, 1], if(issquare(2*n^2+1762*n+776161, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=841, a(2)=881, a(3)=925, a(4)=4121, a(5)=4405, a(6)=4709.

G.f.: (1-x)*(841+1722*x+2647*x^2+1722*x^3+841*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 881*A001653(k) for k >= 1.