cp's OEIS Frontend

A059988 a(n) = (10^n - 1)^2.

%I A059988 #88 Aug 03 2025 16:05:14
%S A059988 0,81,9801,998001,99980001,9999800001,999998000001,99999980000001,
%T A059988 9999999800000001,999999998000000001,99999999980000000001,
%U A059988 9999999999800000000001,999999999998000000000001,99999999999980000000000001,9999999999999800000000000001,999999999999998000000000000001
%N A059988 a(n) = (10^n - 1)^2.
%C A059988 From _James D. Klein_, Feb 05 2012: (Start)
%C A059988 The periods of the reciprocals of a(n) are the consecutive integers from 0 to 10^n-1, omitting the one integer 10^n-2, right-justified in field widths of size n.
%C A059988 E.g.:
%C A059988   1/81 = 0.012345679...
%C A059988   1/9801 = 0.000102030405060708091011...9799000102...
%C A059988   1/998001 = 0.000001002003004005...997999000001002... (End)
%C A059988 Sum of first 10^n - 1 odd numbers. - _Arkadiusz Wesolowski_, Jun 12 2013
%D A059988 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 32 at p. 61.
%D A059988 Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.
%D A059988 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 34.
%H A059988 Harry J. Smith, <a href="/A059988/b059988.txt">Table of n, a(n) for n = 0..200</a>
%H A059988 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A059988 a(n) = 81*A002477(n) = A002283(n)^2 = (9*A002275(n))^2.
%F A059988 a(n) = {999... (n times)}^2 = {999... (n times), 000... (n times)} - {999... (n times)}. For example, 999^2 = 999000 - 999 = 998001. - _Kyle D. Balliet_, Mar 07 2009
%F A059988 a(n) = (A002283(n-1)*10 + 8) * 10^(n-1) + 1, for n>0. - _Reinhard Zumkeller_, May 31 2010
%F A059988 From _Ilya Gutkovskiy_, Apr 19 2016: (Start)
%F A059988 O.g.f.: 81*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
%F A059988 E.g.f.: (1 - 2*exp(9*x) + exp(99*x))*exp(x). (End)
%F A059988 Sum_{n>=1} 1/a(n) = (log(10)*(QPolyGamma(0, 1, 1/10) - log(10/9)) + QPolyGamma(1, 1, 1/10))/log(10)^2 = 0.012448721523422795191... . - _Stefano Spezia_, Jul 31 2024
%F A059988 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3). - _Elmo R. Oliveira_, Aug 02 2025
%e A059988 From _Reinhard Zumkeller_, May 31 2010: (Start)
%e A059988 n=1: ..................... 81 = 9^2;
%e A059988 n=2: ................... 9801 = 99^2;
%e A059988 n=3: ................. 998001 = 999^2;
%e A059988 n=4: ............... 99980001 = 9999^2;
%e A059988 n=5: ............. 9999800001 = 99999^2;
%e A059988 n=6: ........... 999998000001 = 999999^2;
%e A059988 n=7: ......... 99999980000001 = 9999999^2;
%e A059988 n=8: ....... 9999999800000001 = 99999999^2;
%e A059988 n=9: ..... 999999998000000001 = 999999999^2. (End)
%p A059988 A059988:=n->(10^n-1)^2; seq(A059988(n), n=0..20); # _Wesley Ivan Hurt_, Nov 21 2013
%t A059988 Table[(10^n-1)^2, {n,0,20}] (* _Wesley Ivan Hurt_, Nov 21 2013 *)
%o A059988 (PARI) a(n) = { (10^n - 1)^2 } \\ _Harry J. Smith_, Jul 01 2009
%o A059988 (Python) def a(n): return (10**n - 1)**2  # _Martin Gergov_, Sep 10 2022
%Y A059988 Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417, A178630, A178631, A178632, A178633, A178634, A178635, A272066, A272067, A272068.
%Y A059988 Subsequence of A238237.
%Y A059988 Cf. A002275, A002283, A002477.
%K A059988 easy,nonn
%O A059988 0,2
%A A059988 _Henry Bottomley_, Mar 07 2001