This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A168624 #23 Sep 12 2024 19:11:25
%S A168624 1,91,9901,999001,99990001,9999900001,999999000001,99999990000001,
%T A168624 9999999900000001,999999999000000001,99999999990000000001,
%U A168624 9999999999900000000001,999999999999000000000001,99999999999990000000000001,9999999999999900000000000001,999999999999999000000000000001
%N A168624 a(n) = 1 - 10^n + 100^n.
%C A168624 Prime values for n = 2,4,6,8, with no others up to n = 3400. Beiler mentions this pattern in the reference.
%C A168624 From _Peter Bala_, Sep 27 2015: (Start)
%C A168624 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.
%C A168624 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)
%D A168624 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 85.
%H A168624 Colin Barker, <a href="/A168624/b168624.txt">Table of n, a(n) for n = 0..499</a>
%H A168624 P. Bala, <a href="/A168624/a168624.pdf">A168624 and some empirical continued fraction expansions</a>.
%H A168624 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A168624 From _Colin Barker_, Sep 27 2015: (Start)
%F A168624 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
%F A168624 G.f.: -(910*x^2-20*x+1)/((x-1)*(10*x-1)*(100*x-1)). (End)
%F A168624 E.g.f.: exp(x)*(exp(99*x) - exp(9*x) + 1). - _Elmo R. Oliveira_, Sep 12 2024
%e A168624 Simple continued fraction expansions showing large partial quotients:
%e A168624 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, ...].
%e A168624 a(18)^(1/3) = [999999999999; 1, 2999999, 499999999999, 1, 1439999, 2582644628099, 5, 1, 3, 4, 1, 58, 1, 1, 1, 8, ...].
%e A168624 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
%t A168624 Table[1-10^n+100^n,{n,0,20}] (* _Harvey P. Dale_, Dec 01 2013 *)
%o A168624 (PARI) Vec(-(910*x^2-20*x+1)/((x-1)*(10*x-1)*(100*x-1)) + O(x^20)) \\ _Colin Barker_, Sep 27 2015
%Y A168624 Cf. A000533, A002283, A066138, A187868.
%K A168624 easy,nonn
%O A168624 0,2
%A A168624 _Jason Earls_, Dec 01 2009