A053422 n times (n 1's): a(n) = n*(10^n - 1)/9.

0, 1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 11111111110, 122222222221, 1333333333332, 14444444444443, 155555555555554, 1666666666666665, 17777777777777776, 188888888888888887, 1999999999999999998, 21111111111111111109, 222222222222222222220, 2333333333333333333331

Offset: 0

Author

Henry Bottomley, Mar 07 2000

Keywords

• base
• easy
• nonn

Comments

R_a(n) is the least repunit divisible by the square of R_n = (10^n - 1)/9. - Lekraj Beedassy, Jun 07 2006

Links

• G. C. Greubel, Table of n, a(n) for n = 0..995
• Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100).

Crossrefs

Cf. A000461, A002275, A048376.

Programs

Magma

I:=[0, 1, 22, 333]; [n le 4 select I[n] else 22*Self(n-1) - 141*Self(n-2) +220*Self(n-3) -100*Self(n-4): n in [1..30]]; // G. C. Greubel, May 25 2018

Mathematica

LinearRecurrence[{22,-141,220,-100}, {0, 1, 22, 333}, 50] (* G. C. Greubel, May 25 2018 *)
CoefficientList[Series[x (1-10x^2)/((1-x)^2(1-10x)^2),{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)

PARI

x='x+O('x^30); concat([0], Vec(x*(1-10*x^2)/((1-x)^2*(1-10*x)^2))) \\ G. C. Greubel, May 25 2018

Sage

[gaussian_binomial(n,1,10)*n for n in range(0,22)] # Zerinvary Lajos, May 29 2009

Formula

a(n) = n*A002275(n) = a(n-1)*10n/(n-1) + n.

O.g.f.: x*(1-10*x^2)/((1-x)^2*(1-10*x)^2). - R. J. Mathar, Jan 21 2008

E.g.f.: x*exp(x)*(10*exp(9*x) - 1)/9. - Stefano Spezia, Sep 14 2023

Extensions

Corrected by Jason Earls, Sep 02 2006