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A000042 Unary representation of natural numbers.
(Formerly M4804)
70
1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Or, numbers written in base 1.

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, Oct 05 2002

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. [Vincenzo Librandi, Jul 18 2010]

REFERENCES

K. G. Kroeber, Mathematik der Palindrome; p. 348; 2003; ISBN 3 499 615762; Rowohlt Verlag; Germany

Amarnath Murthy, On the divisors of the unary sequence, Smarandache Notions Journal Vol. - 11, 2000.

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.

D. Olivastro, Ancient Puzzles. Bantam Books, NY, 1993, p. 276.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

David Wasserman, Table of n, a(n) for n=1..1000

Index to sequences with linear recurrences with constant coefficients, signature (11,-10).

Index to divisibility sequences

FORMULA

a(n) = (10^n-1)/9.

G.f.: 1/((1-x)*(1-10*x)).

Binomial transform of A003952. - Paul Barry, Jan 29 2004

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, Aug 24 2004

a(2n) -2*a(n) ={3*a(n)}^2. a(6)-2*a(3) = {3*a(3)}^2. 111111-222 = 110889 - Amarnath Murthy, Jul 21 2003

a(n) = binary representation of n-th Mersenne number (A000225). - Ross La Haye, Sep 13 2003

The Hankel transform of this sequence is [1,-10,0,0,0,0,0,0,0,0,...] - Philippe Deléham, Nov 21 2007

MATHEMATICA

Table[(10^n - 1)/9, {n, 1, 18}]

FromDigits/@Table[PadLeft[{}, n, 1], {n, 20}] (* Harvey P. Dale, Aug 21 2011 *)

PROG

(PARI) a(n)=if(n<0, 0, (10^n-1)/9)

(Sage) [gaussian_binomial(n, 1, 10) for n in xrange(1, 19)] # [Zerinvary Lajos, May 28 2009]

(Haskell) A000042 n = (10^n-1) `div` 9 -- James Spahlinger, Oct 08 2012

(Common Lisp) (defun a000042 (n) (truncate (expt 10 n) 9)) ; James Spahlinger, Oct 12 2012

CROSSREFS

Cf. A002275, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095.

Sequence in context: A165370 A134962 A113589 * A135463 A002275 A078998

Adjacent sequences:  A000039 A000040 A000041 * A000043 A000044 A000045

KEYWORD

base,easy,nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Paul Barry, Jan 29 2004

STATUS

approved

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Last modified October 30 12:40 EDT 2014. Contains 248801 sequences.