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) = 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
(SageMath)
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
KEYWORD
nonn,easy
AUTHOR
Gregory V. Richardson, Oct 14 2002
STATUS
approved