OFFSET
0,2
COMMENTS
Prime values for n = 2,4,6,8, with no others up to n = 3400. Beiler mentions this pattern in the reference.
From Peter Bala, Sep 27 2015: (Start)
Calculation suggests the continued fraction expansion of sqrt(a(n)), for n >= 1, begins [10^n - 1, 1, 1, 1/3*(2*10^n - 5), 1, 5, 1/9*(2*10^n - 11), 1, 17, (2*10^n - 20 - 9*(1 - MOD(n, 3)))/27, ...]. Note the large partial quotients early in the expansion.
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions. Empirically, we also see exceptionally large partial quotients in the continued fraction expansions of the m-th root of the numbers a(m*n) for m = 2, 3, 4,... as n increases. Some examples are given below. Cf. A000533, A002283, A066138. (End)
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 85.
LINKS
Colin Barker, Table of n, a(n) for n = 0..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
From Colin Barker, Sep 27 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
G.f.: -(910*x^2-20*x+1)/((x-1)*(10*x-1)*(100*x-1)). (End)
E.g.f.: exp(x)*(exp(99*x) - exp(9*x) + 1). - Elmo R. Oliveira, Sep 12 2024
EXAMPLE
Simple continued fraction expansions showing large partial quotients:
sqrt(a(10)) = [9999999999; 1, 1, 6666666665, 1, 5, 2222222221, 1, 17, 740740740, 1, 1, 1, 5, 2, 1, 246913579, 1, 1, 4, 1, 1, 3, 1, 1, ...].
a(18)^(1/3) = [999999999999; 1, 2999999, 499999999999, 1, 1439999, 2582644628099, 5, 1, 3, 4, 1, 58, 1, 1, 1, 8, ...].
a(30)^(1/5) = [999999999999; 1, 4999999999999999999, 333333333333, 3, 217391304347826086, 1, 1, 1, 1, 1, 8, 2398081534, 1, 1, 1, 9, 1, 98, 1, 125052522059263, 1, 9, 7, 1, ...]. - Peter Bala, Sep 27 2015
MATHEMATICA
Table[1-10^n+100^n, {n, 0, 20}] (* Harvey P. Dale, Dec 01 2013 *)
PROG
(PARI) Vec(-(910*x^2-20*x+1)/((x-1)*(10*x-1)*(100*x-1)) + O(x^20)) \\ Colin Barker, Sep 27 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Dec 01 2009
STATUS
approved