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A002275
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Repunits: (10^n - 1)/9. Often denoted by R_n.
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794
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0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111
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OFFSET
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0,3
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COMMENTS
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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]
These numbers are particularly helpful when translating permutations between two different kind of conventions about the enumerations of the digits. Example: {01,10} + {11,11} = {12,21}. - R. J. Cano, Jan 09 2013
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REFERENCES
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Snyder, W. M. "Factoring Repunits." Am. Math. Monthly 89, 462-466, 1982.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 197-8 Penguin Books 1987.
Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139-146, 1969.
Yates, S. "Prime Divisors of Repunits." J. Recr. Math. 8, 33-38, 1975.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.
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LINKS
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David Wasserman, Table of n, a(n) for n=0..1000
Index entries for sequences related to linear recurrences with constant coefficients
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Demlo Number
Wikipedia, Repunit
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=(10^n-1)/9; [From Michael B. Porter, Oct 26 2009]
(Sage) [lucas_number1(n, 11, 10) for n in xrange(0, 21)]# [Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 27 2009]
(Haskell)
a002275 n = a002275_list !! n
a002275_list = iterate ((+ 1) . (* 10)) 0
-- Reinhard Zumkeller, 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 */
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CROSSREFS
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Cf. A000042. Partial sums of 10^n (A011557). Factors: A003020, A067063.
Bisections give A099814, A100706.
Cf. A046053, A095370.
Cf. A002276, A002277, A002278, A002279, A002280, A002281, A002282, A059988, A075415, A178635. [From Reinhard Zumkeller, May 31 2010]
Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).
Cf. A102380, A204845, A204846, A204847, A204848.
Cf. A083278, A206244.
Cf. A125134, A004023.
Sequence in context: A113589 A000042 A135463 * A078998 A078191 A097115
Adjacent sequences: A002272 A002273 A002274 * A002276 A002277 A002278
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KEYWORD
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easy,nonn,nice,core
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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