A168624 a(n) = 1 - 10^n + 100^n.

1, 91, 9901, 999001, 99990001, 9999900001, 999999000001, 99999990000001, 9999999900000001, 999999999000000001, 99999999990000000001, 9999999999900000000001, 999999999999000000000001, 99999999999990000000000001, 9999999999999900000000000001, 999999999999999000000000000001

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

P. Bala, A168624 and some empirical continued fraction expansions.

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

Cf. A000533, A002283, A066138, A187868.

Sequence in context: A165154 A015261 A370780 * A131442 A319370 A116507

Adjacent sequences: A168621 A168622 A168623 * A168625 A168626 A168627

Keyword

easy,nonn

Author

Jason Earls, Dec 01 2009

Status

approved