%I A000042 M4804
%S A000042 1,11,111,1111,11111,111111,1111111,11111111,111111111,1111111111,
%T A000042 11111111111,111111111111,1111111111111,11111111111111,111111111111111,
%U A000042 1111111111111111,11111111111111111,111111111111111111
%N A000042 Unary representation of natural numbers.
%C A000042 Or, numbers written in base 1.
%C A000042 If p is a prime >5 then d_{a(p)} == 1 mod (p) where d_{a(p)} is a divisor 
               of a(p). This also gives an alternate elementary proof of the infinitude 
               of prime numbers by the fact that for every prime p there exists 
               at least one prime of the form kp+1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Oct 05 2002
%C A000042 11=1*9+2; 111=12*9+3; 1111=123*9+4; 11111=1234*9+5; 111111=12345*9+6; 
               1111111=123456*9+7; 11111111=1234567*9+8; 111111111=12345678*9+9. 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 17 2009]
%D A000042 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000042 K. G. Kroeber, Mathematik der Palindrome; p. 348; 2003; ISBN 3 499 615762; 
               Rowohlt Verlag; Germany
%D A000042 D. Olivastro, Ancient Puzzles. Bantam Books, NY, 1993, p. 276.
%D A000042 Amarnath Murthy, On the divisors of the unary sequence, Smarandache Notions 
               Journal Vol. - 11, 2000.
%D A000042 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some 
               New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; 
               USA 2005. See Section 2.12.
%H A000042 David Wasserman, <a href="b000042.txt">Table of n, a(n) for n=1..1000</
               a>
%H A000042 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000042 a(n) = (10^n-1)/9.
%F A000042 G.f.: 1/((1-x)*(1-10*x)).
%F A000042 Binomial transform of A003952. - Paul Barry (pbarry(AT)wit.ie), Jan 29 
               2004
%F A000042 a(n)=10a(n-1)+1, n>1, a(1)=1. [Offset 1]. a(n)=sum{k=0..n, binomial(n+1, 
               k+1)9^k}. [Offset 0]. - Paul Barry (pbarry(AT)wit.ie), Aug 24 2004
%F A000042 a(2n) -2*a(n) ={3*a(n)}^2. a(6)-2*a(3) = {3*a(3)}^2. 111111-222 = 110889 
               - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 21 2003
%F A000042 a(n) = binary representation of n-th Mersenne number (A000225). - Ross 
               La Haye (rlahaye(AT)new.rr.com), Sep 13 2003
%F A000042 The Hankel transform of this sequence is [1,-10,0,0,0,0,0,0,0,0,...] 
               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
%t A000042 Table[(10^n - 1)/9, {n, 1, 18}]
%o A000042 (PARI) a(n)=if(n<0,0,(10^n-1)/9)
%o A000042 (Other) sage: [gaussian_binomial(n,1,10) for n in xrange(1,19)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
%Y A000042 Cf. A002275, A007088, A007089, A007090, A007091, A007092, A007093, A007094 
               & A007095.
%Y A000042 Sequence in context: A165370 A134962 A113589 this_sequence A135463 A002275 
               A078998
%Y A000042 Adjacent sequences: A000039 A000040 A000041 this_sequence A000043 A000044 
               A000045
%K A000042 easy,nonn,nice
%O A000042 1,2
%A A000042 N. J. A. Sloane (njas(AT)research.att.com).
%E A000042 More terms from Paul Barry (pbarry(AT)wit.ie), Jan 29 2004