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 A168624 a(n) = 1 - 10^n + 100^n. 8
 1, 91, 9901, 999001, 99990001, 9999900001, 999999000001, 99999990000001, 9999999900000001, 999999999000000001, 99999999990000000001, 9999999999900000000001, 999999999999000000000001 (list; graph; refs; listen; history; text; internal format)
 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) 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. A187868, A000533, A002283, A066138. Sequence in context: A095372 A165154 A015261 * A131442 A319370 A116507 Adjacent sequences:  A168621 A168622 A168623 * A168625 A168626 A168627 KEYWORD easy,nonn AUTHOR Jason Earls, Dec 01 2009 STATUS approved

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Last modified May 14 07:13 EDT 2021. Contains 343879 sequences. (Running on oeis4.)