A075836 Numbers k such that 10*k^2 + 9 is a square.

0, 2, 4, 18, 80, 154, 684, 3038, 5848, 25974, 115364, 222070, 986328, 4380794, 8432812, 37454490, 166354808, 320224786, 1422284292, 6317101910, 12160109056, 54009348606, 239883517772, 461763919342, 2050932962736

Offset

1,2

Comments

(5/4)*a(n)^2 +1 is a triangular number. - Bruno Berselli, Aug 17 2013

References

A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.

Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

J. J. O'Connor and E. F. Robertson, Pell's Equation

Eric Weisstein's World of Mathematics, Pell Equation.

Index entries for linear recurrences with constant coefficients, signature (0,0,38,0,0,-1).

Formula

From Gregory V. Richardson, Oct 16 2002: (Start)

Limit_{n->oo} a(n)/a(n-3) = 19 + 6*sqrt(10).

Limit_{n->oo} a(3*n)/a(3*n-1) = (11 + 2*sqrt(10))/9.

Limit_{n->oo} a(3*n+1)/a(3*n) = (7 + 2*sqrt(10))/3.

Limit_{n->oo} a(3*n+2)/a(3*n+1) = (7 + 2*sqrt(10))/3. (End)

G.f.: 2*x^2*(1+2*x+9*x^2+2*x^3+x^4) / ( 1-38*x^3+x^6 ). - R. J. Mathar, Jul 03 2011

a(n) = 2*A075873(n). - R. J. Mathar, Jul 03 2011

Mathematica

CoefficientList[Series[2 x (1 + 2 x + 9 x^2 + 2 x^3 + x^4) / (1 - 38 x^3 + x^6), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 17 2013 *)

Prog

(Magma) I:=[0, 2, 4, 18, 80, 154]; [n le 6 select I[n] else 38*Self(n-3)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 17 2013

Crossrefs

Cf. A221874.

Sequence in context: A277033 A295767 A318230 * A295370 A292280 A120664

Adjacent sequences: A075833 A075834 A075835 * A075837 A075838 A075839

Keyword

nonn,easy

Author

Gregory V. Richardson, Oct 14 2002

Status

approved