A075844 Numbers k such that 11*k^2 + 4 is a square.

0, 6, 120, 2394, 47760, 952806, 19008360, 379214394, 7565279520, 150926376006, 3010962240600, 60068318435994, 1198355406479280, 23907039811149606, 476942440816512840, 9514941776519107194, 189821893089565631040

Offset: 0

Author

Gregory V. Richardson, Oct 14 2002

Keywords

• nonn
• easy

Comments

Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*sqrt(11).

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

• Harvey P. Dale, Table of n, a(n) for n = 0..750
• Tanya Khovanova, Recursive Sequences
• 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 (20,-1).

Crossrefs

Cf. A221762.

Programs

GAP

a:=[0,6];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019

Magma

R:=PowerSeriesRing(Integers(), 20); [0] cat Coefficients(R!( 6*x/(1 - 20*x + x^2) )); // G. C. Greubel, Dec 06 2019

Maple

seq(coeff(series(6*x/(1-20*x+x^2), x, n+1), x, n), n = 0..20); # G. C. Greubel, Dec 06 2019

Mathematica

LinearRecurrence[{20,-1},{0,6},20] (* Harvey P. Dale, May 28 2012 *)

PARI

my(x='x+O('x^20)); concat([0], Vec(6*x/(1-20*x+x^2))) \\ G. C. Greubel, Dec 06 2019

Sage

def A075844_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 6*x/(1-20*x+x^2) ).list()
A075844_list(20) # G. C. Greubel, Dec 06 2019

Formula

a(n) = ((10+3*sqrt(11))^n - (10-3*sqrt(11))^n) / sqrt(11).

a(n) = 20*a(n-1) - a(n-2).

G.f.: 6*x/(1 - 20*x + x^2).

a(n) = (1/3)*(A075839(n+1) - A075839(n)), n>=1. - N. J. A. Sloane, Sep 22 2004

a(n) = 6*A075843(n). - R. J. Mathar, Jul 03 2011