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 A002283 a(n) = 10^n - 1. 168
 0, 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 999999999999999, 9999999999999999, 99999999999999999, 999999999999999999, 9999999999999999999, 99999999999999999999, 999999999999999999999, 9999999999999999999999 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A friend from Germany remarks that the sequence 9, 99, 999, 9999, 99999, 999999, ... might be called the grumpy German sequence: nein!, nein! nein!, nein! nein! nein!, ... The Regan link shows that integers of the form 10^n -1 have binary representations with exactly n trailing 1 bits. Also those integers have quinary expressions with exactly n trailing 4's. For example, 10^4 -1 = (304444)5. The first digits in quinary correspond to the number 2^n -1, in our example (30)5 = 2^4 -1. A similar pattern occurs in the binary case. Consider 9 = (1001)2. - Washington Bomfim Dec 23 2010 a(n) is the number of positive integers with less than n+1 digits. - Bui Quang Tuan, Mar 09 2015 From Peter Bala, Sep 27 2015: (Start) For n >= 1, the simple continued fraction expansion of sqrt(a(2*n)) = [10^n - 1; 1, 2*(10^n - 1), 1, 2*(10^n - 1), ...] has period 2. The simple continued fraction expansion of sqrt(a(2*n))/a(n) = [1; 10^n - 1, 2,  10^n - 1, 2, ...] also has period 2. Note the occurrence of large partial quotients in both expansions. 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 the presence of unexpectedly large partial quotients early in the continued fraction expansions of the m-th roots of the numbers a(m*n) for m >= 3. Some typical examples are given below. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 Rick Regan, Nines in quinary, September 8th, 2009. Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT]. Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10. Index entries for linear recurrences with constant coefficients, signature (11,-10). FORMULA G.f.: 1/(1-10*x)-1/(1-x). E.g.f.: e^(10*x)-e^x. - Mohammad K. Azarian, Jan 14 2009 a(n) = 10*a(n-1) + 9, with a(0)=0. - Paolo P. Lava, Jan 23 2009 a(n) = A075412(n)/A002275(n) = A178630(n)/A002276(n) = A178631(n)/A002277(n) = A075415(n)/A002278(n) = A178632(n)/A002279(n) = A178633(n)/A002280(n) = A178634(n)/A002281(n) = A178635(n)/A002282(n). - Reinhard Zumkeller, May 31 2010 a(n) = a(n-1) + 9*10^(n-1) with a(0)=0; Also: a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=9. - Vincenzo Librandi, Jul 22 2010 For n>0, A007953(a(n)) = A008591(n) and A010888(a(n)) = 9. - Reinhard Zumkeller, Aug 06 2010 A048379(a(n)) = 0. - Reinhard Zumkeller, Feb 21 2014 a(n) = Sum_{k=1..n} 9*10^k. - Carauleanu Marc, Sep 03 2016 EXAMPLE From Peter Bala, Sep 27 2015: (Start) Continued fraction expansions showing large partial quotients: a(12)^(1/3) = [9999; 1, 299999998, 1, 9998, 1, 449999998, 1, 7998, 1, 535714284, 1, 2, 2, 142, 2, 2, 1, 599999999, 3, 1, 1,...]. Compare with a(30)^(1/3) = [9999999999; 1, 299999999999999999998, 1, 9999999998, 1, 449999999999999999998, 1, 7999999998, 1, 535714285714285714284, 1, 2, 2, 142857142, 2, 2, 1, 599999999999999999999, 3, 1, 1,...]. a(24)^(1/4) = [999999; 1, 3999999999999999998, 1, 666665, 1, 1, 1, 799999999999999999, 3, 476190, 7, 190476190476190476, 21, 43289, 1, 229, 1, 1864801864801863, 1, 4, 6,...]. Compare with a(48)^(1/4) = [999999999999; 1, 3999999999999999999999999999999999998, 1, 666666666665, 1, 1, 1, 799999999999999999999999999999999999, 3, 476190476190, 7, 190476190476190476190476190476190476, 21, 43290043289, 1, 229, 1, 1864801864801864801864801864801863, 1, 4, 6,...]. a(25)^(1/5) = [99999, 1, 499999999999999999998, 1, 49998, 1, 999999999999999999998, 1, 33332, 3, 151515151515151515151, 5, 1, 1, 1947, 1, 1, 38, 3787878787878787878, 1, 3, 5,...]. (End) MATHEMATICA Table[10^n - 1, {n, 0, 22}] (* Michael De Vlieger, Sep 27 2015 *) PROG (MAGMA) [(10^n-1): n in [0..20]]; // Vincenzo Librandi, Apr 26 2011 (PARI) a(n)=10^n-1; \\ Charles R Greathouse IV, Jan 30 2012 (Maxima) A002283(n):=10^n-1\$ makelist(A002283(n), n, 0, 20); /* Martin Ettl, Nov 08 2012 */ (Haskell) a002283 = subtract 1 . (10 ^)  -- Reinhard Zumkeller, Feb 21 2014 CROSSREFS Cf. A000533, A003020, A007138, A066138, A168624, A276352. Sequence in context: A108908 A116260 A103456 * A155157 A264005 A232943 Adjacent sequences:  A002280 A002281 A002282 * A002284 A002285 A002286 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Michael De Vlieger, Sep 27 2015 STATUS approved

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Last modified October 26 03:21 EDT 2020. Contains 338027 sequences. (Running on oeis4.)