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

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111

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 >= 0: a(n) = (A000225(n) written in base 2). - Jaroslav Krizek, Jul 27 2009, edited by M. F. Hasler, Jul 03 2020

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

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

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

From Peter Bala, Sep 20 2015: (Start)

Let d be a divisor of a(n). Let m*d be any multiple of d. Split the decimal expansion of m*d into 2 blocks of contiguous digits a and b, so we have m*d = 10^k*a + b for some k, where 0 <= k < number of decimal digits of m*d. Then d divides a^n - (-b)^n (see McGough). For example, 271 divides a(5) and we find 2^5 + 71^5 = 11*73*271*8291 and 27^5 + 1^5 = 2^2*7*31*61*271 are both divisible by 271. Similarly, 4*271 = 1084 and 10^5 + 84^5 = 2^5*31*47*271*331 while 108^5 + 4^5 = 2^12*7*31*61*271 are again both divisible by 271. (End)

Starting with the second term this sequence is the binary representation of the n-th iteration of the Rule 220 and 252 elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016

If p > 5 is a prime, then p divides a(p-1). - Thomas Ordowski, Apr 10 2016

0, 1 and 11 are only terms that are of the form x^2 + y^2 + z^2 where x, y, z are integers. In other words, a(n) is a member of A004215 for all n > 2. - Altug Alkan, May 08 2016

Except for the initial terms, the binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 17 2017

The term "repunit" was coined by Albert H. Beiler in 1964. - Amiram Eldar, Nov 13 2020

q-integers for q = 10. - John Keith, Apr 12 2021

Binomial transform of A001019 with leading zero. - Jules Beauchamp, Jan 04 2022

References

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, chapter XI, p. 83.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 197-198.

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 11...11 (Repunit).

W. M. Snyder, Factoring Repunits, Am. Math. Monthly, Vol. 89, No. 7 (1982), pp. 462-466.

Amelia Carolina Sparavigna, On Repunits, Politecnico di Torino (Italy, 2019).

Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.

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.

Eric Weisstein's World of Mathematics, Repunit.

Eric Weisstein's World of Mathematics, Demlo Number.

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.

Wikipedia, Repunit.

Amin Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6.

Stephen Wolfram, A New Kind of Science.

Samuel Yates, The Mystique of Repunits, Math. Mag., Vol. 51, No. 1 (1978), pp. 22-28.

Index to Elementary Cellular Automata

Index entries for 10-automatic sequences.

Index entries for sequences related to cellular automata

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Index entries for "core" sequences

Formula

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

a(n) = A000042(n).

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

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

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

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

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

E.g.f.: (exp(9*x) - 1)*exp(x)/9. - Ilya Gutkovskiy, May 11 2016

a(n) = Sum_{k=0..n-1} 10^k. - Torlach Rush, Nov 03 2020

Sum_{n>=1} 1/a(n) = A065444. - Amiram Eldar, Nov 13 2020

Maple

seq((10^k - 1)/9, k=0..30); # 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
(PARI) x='x+O('x^99); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ Altug Alkan, Apr 10 2016
(Sage) [lucas_number1(n, 11, 10) for n in range(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
(Python)
print([(10**n-1)//9 for n in range(100)]) # Michael S. Branicky, Apr 30 2022

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, A065444, 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 A135463 A000042 * A294348 A078998 A078191

Adjacent sequences: A002272 A002273 A002274 * A002276 A002277 A002278

Keyword

easy,nonn,nice,core,changed

Author

N. J. A. Sloane

Status

approved