A016123 a(n) = (11^(n+1) - 1)/10.

1, 12, 133, 1464, 16105, 177156, 1948717, 21435888, 235794769, 2593742460, 28531167061, 313842837672, 3452271214393, 37974983358324, 417724816941565, 4594972986357216, 50544702849929377, 555991731349223148

Offset

0,2

Comments

11^a(n) is highest power of 11 dividing (11^(n+1))!.

Partial sums of powers of 11 (A001020).

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=11, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det(A). - Milan Janjic, Feb 21 2010

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=12, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010

Links

G. C. Greubel, Table of n, a(n) for n = 0..950

Eric Weisstein's World of Mathematics, Repunit

Index entries for linear recurrences with constant coefficients, signature (12,-11).

Formula

a(n) = Sum_{k=0..n} 11^k = (11^(n+1) - 1)/10.

G.f.: (1/(1-11*x) - 1/(1-x))/(10*x) = 1/((1-11*x)*(1-x)).

a(0)=1, a(n) = 11*a(n-1) + 1. - Vincenzo Librandi, Feb 05 2011

a(0)=0, a(1)=1, a(n) = 12*a(n-1) - 11*a(n-2). - Harvey P. Dale, Apr 05 2012

E.g.f.: exp(x)*(11*exp(10*x) - 1)/10. - Stefano Spezia, Mar 11 2023

Mathematica

(11^Range[0, 20]-1)/10 (* or *) LinearRecurrence[{12, -11}, {0, 1}, 20] (* Harvey P. Dale, Apr 05 2012 *)

Prog

(Sage) [lucas_number1(n, 12, 11) for n in range(1, 19)] # Zerinvary Lajos, Apr 27 2009
(Sage) [gaussian_binomial(n, 1, 11) for n in range(1, 19)] # Zerinvary Lajos, May 28 2009
(Maxima) A016123(n):=(11^(n+1)-1)/10$
makelist(A016123(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(PARI) a(n)=(11^(n+1)-1)/10 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [(11^(n+1)-1)/10: n in [0..30]]; // G. C. Greubel, Feb 21 2024

Crossrefs

For analogs with primes 2, 3, 5, 7, 13 and 17 see: A000225, A003462, A003463, A023000, A091030 and A091045, respectively.

Cf. A001020, A004191.

Sequence in context: A120673 A120674 A244205 * A353148 A015457 A015469

Adjacent sequences: A016120 A016121 A016122 * A016124 A016125 A016126

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Title edited by Daniel Forgues, Jul 08 2011

Status

approved