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A000042 Unary representation of natural numbers.
(Formerly M4804)
97
1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111 (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 k*p + 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.
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
Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000, page 184.
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.
FORMULA
a(n) = (10^n - 1)/9.
G.f.: 1/((1-x)*(1-10*x)).
Binomial transform of A003952. - Paul Barry, Jan 29 2004
From Paul Barry, Aug 24 2004: (Start)
a(n) = 10*a(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.] (End)
a(2n) - 2*a(n) = (3*a(n))^2. - Amarnath Murthy, Jul 21 2003
a(n) is the binary representation of the 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
E.g.f.: (exp(10*x) - exp(x))/9. - G. C. Greubel, Nov 04 2018
a(n) = 11*a(n-1) - 10*a(n-2). - Wesley Ivan Hurt, May 28 2021
a(n+m-2) = a(m)*a(n-1) - (a(m)-1)*a(n-2), n>1, m>0. - Matej Veselovac, Jun 07 2021
MAPLE
a:= n-> parse(cat(1$n)):
seq(a(n), n=1..25); # Alois P. Heinz, Mar 23 2018
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 range(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
(Magma) [(10^n - 1)/9: n in [1..20]]; // G. C. Greubel, Nov 04 2018
(Python) def a(n): return int("1"*n) # Michael S. Branicky, Jan 01 2021
CROSSREFS
Sequence in context: A113589 A366394 A135463 * A002275 A294348 A078998
KEYWORD
base,easy,nonn,nice
AUTHOR
EXTENSIONS
More terms from Paul Barry, Jan 29 2004
STATUS
approved

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Last modified February 20 20:53 EST 2024. Contains 370217 sequences. (Running on oeis4.)