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A075839 11*n^2 - 2 is a square. 11
1, 19, 379, 7561, 150841, 3009259, 60034339, 1197677521, 23893516081, 476672644099, 9509559365899, 189714514673881, 3784780734111721, 75505900167560539, 1506333222617099059, 30051158552174420641, 599516837820871313761, 11960285597865251854579 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Positive values of x (or y) satisfying x^2 - 20xy + y^2 + 18 = 0. - Colin Barker, Feb 18 2014

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 = 0..200

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 two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (20,-1).

FORMULA

11*a(n)^2-9*A083043(n)^2=2.

a(n) = ((3+sqrt(11))*(10+3*sqrt(11))^n - (3-sqrt(11))*(10-3*sqrt(11))^n)/(2*sqrt(11)). - Dean Hickerson, Dec 09 2002

G.f.: (1-x)/(1-20*x+x^2). a(n)=20*a(n-1)-a(n-2), n>1. - Michael Somos, Oct 29 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)) then a(n)=q(n, 18). - Benoit Cloitre, Dec 06 2002

a(-1-n)=a(n). - Michael Somos, Apr 18 2003

MATHEMATICA

LinearRecurrence[{20, -1}, {1, 19}, 20] (* Harvey P. Dale, Apr 13 2012 *)

CoefficientList[Series[(1 - x)/(1 - 20 x + x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 20 2014 *)

a[c_, n_] := Module[{},

   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

   d := Denominator[Convergents[Sqrt[c], n p]];

   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

   Return[t];

] (* Complement of A041015 *)

a[11, 20] (* Gerry Martens, Jun 07 2015 *)

PROG

(PARI) a(n)=subst(poltchebi(n+1)+poltchebi(n), x, 10)/11

(MAGMA) I:=[1, 19]; [n le 2 select I[n] else 20*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 20 2014

CROSSREFS

Row 20 of array A094954.

Cf. A075844, A221762, A041015.

Cf. similar sequences listed in A238379.

Sequence in context: A041686 A263371 A023283 * A158592 A072359 A222835

Adjacent sequences:  A075836 A075837 A075838 * A075840 A075841 A075842

KEYWORD

easy,nonn

AUTHOR

Gregory V. Richardson, Oct 14 2002

EXTENSIONS

More terms from Colin Barker, Feb 18 2014

STATUS

approved

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Last modified November 19 00:50 EST 2017. Contains 294912 sequences.