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A075844 Numbers n such that 11*n^2 + 4 is a square. 3
0, 6, 120, 2394, 47760, 952806, 19008360, 379214394, 7565279520, 150926376006, 3010962240600, 60068318435994, 1198355406479280, 23907039811149606, 476942440816512840, 9514941776519107194, 189821893089565631040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

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 *)

PROG

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

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

(Sage)

def A075844_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( 6*x/(1-20*x+x^2) ).list()

A075844_list(20) # G. C. Greubel, Dec 06 2019

(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

CROSSREFS

Cf. A221762.

Sequence in context: A246191 A246611 A185757 * A029697 A248045 A280627

Adjacent sequences:  A075841 A075842 A075843 * A075845 A075846 A075847

KEYWORD

nonn,easy

AUTHOR

Gregory V. Richardson, Oct 14 2002

STATUS

approved

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Last modified July 8 22:01 EDT 2020. Contains 335537 sequences. (Running on oeis4.)