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A002275 Repunits: (10^n - 1)/9. Often denoted by R_n. 816
0, 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

0,3

COMMENTS

R_n is a string of n 1's.

Base 4 representation of Jacobsthal bisection sequence A002450. E.g., a(4)= 1111 because A002450(4)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1*(4^3)+1*(4^2)+1*(4^1)+1. - Paul Barry, Mar 12 2004

Except for the first two terms, these numbers cannot be perfect squares, because x^2 =/= 11 (mod 100). - Zak Seidov, Dec 05 2008

For n >= 2: a(n) = Sequence A000225(n) written in base 2. - Jaroslav Krizek, Jul 27 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010

a(n) = A075412(n)/A002283(n). - Reinhard Zumkeller, May 31 2010

Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - Bernard Schott, Dec 24 2012

Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - Bernard Schott, Dec 24 2012

The terms 0 and 1 are the only squares in this sequence, as a(n) == 3 (mod 4) for n>=2. - Nehul Yadav, Sep 26 2013

a(n) = A242614(n,A242622(n)). - Reinhard Zumkeller, Jul 17 2014

For n>=2 the multiplicative order of 10 modulo the a(n) is n. - Robert G. Wilson v, Aug 20 2014

The above is a special case of the statement that the order of z modulo (z^n-1)/(z-1) is n, here for z=10. - Joerg Arndt, Aug 21 2014

REFERENCES

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 197-8 Penguin Books 1987.

Samuel Yates, Peculiar Properties of Repunits, J. Recr. Math. 2, 139-146, 1969.

Samuel Yates, Prime Divisors of Repunits, J. Recr. Math. 8, 33-38, 1975.

LINKS

David Wasserman, Table of n, a(n) for n = 0..1000

Makoto Kamada, Factorizations of the repunits

W. M. Snyder, Factoring Repunits, Am. Math. Monthly 89, 462-466, 1982.

Eric Weisstein's World of Mathematics, Repunit

Eric Weisstein's World of Mathematics, Demlo Number

Wikipedia, Repunit

Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

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

FORMULA

G.f.: x/((1-10*x)*(1-x)). Regarded as base b numbers, g.f. x/((1-b*x)*(1-x)). - Franklin T. Adams-Watters, Jun 15 2006

a(n) = 11*a(n-1)-10*a(n-2), a(0)=0, a(1)=1. - Lekraj Beedassy, Jun 07 2006

a(n) = 10*a(n-1)+1, a(0)=0.

a(n) = a(n-1)+10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010

Second binomial transform of Jacobsthal trisection A001045(3n)/3 (A015565). - Paul Barry, Mar 24 2004

a(n) = A125118(n,9) for n>8. - Reinhard Zumkeller, Nov 21 2006

MAPLE

seq(10^k - 1)/9, k=0..100); # Wesley Ivan Hurt, Sep 28 2013

MATHEMATICA

Table[(10^n - 1)/9, {n, 0, 19}] (* Alonso del Arte, Nov 15 2011 *)

Join[{0}, Table[FromDigits[PadRight[{}, n, 1]], {n, 20}]] (* Harvey P. Dale, Mar 04 2012 *)

PROG

(PARI) a(n)=(10^n-1)/9; \\ Michael B. Porter, Oct 26 2009

(Sage) [lucas_number1(n, 11, 10) for n in xrange(0, 21)] # Zerinvary Lajos, Apr 27 2009

(Haskell)

a002275 = (`div` 9) . subtract 1 . (10 ^)

a002275_list = iterate ((+ 1) . (* 10)) 0

-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

(Maxima)

a[0]:0$

a[1]:1$

a[n]:=11*a[n-1]-10*a[n-2]$

A002275(n):=a[n]$

makelist(A002275(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

(MAGMA) [(10^n-1)/9: n in [0..25]]; // Vincenzo Librandi, Nov 06 2014

CROSSREFS

Partial sums of 10^n (A011557). Factors: A003020, A067063.

Bisections give A099814, A100706.

Cf. A000042, A046053, A095370, A002276, A002277, A002278, A002279, A002280, A002281, A002282, A059988, A075415, A178635, A102380, A204845, A204846, A204847, A204848, A083278, A206244, A125134, A004023.

Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.

Sequence in context: A113589 A000042 A135463 * A078998 A078191 A097115

Adjacent sequences:  A002272 A002273 A002274 * A002276 A002277 A002278

KEYWORD

easy,nonn,nice,core

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 26 01:49 EST 2014. Contains 250017 sequences.