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

1,1

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.

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).

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.

Example

(-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.

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 *)

Prog

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

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).

Sequence in context: A252780 A364264 A362440 * A210470 A108324 A133496

Adjacent sequences: A159687 A159688 A159689 * A159691 A159692 A159693

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 21 2009

Status

approved