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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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
• D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
• Eric Weisstein's World of Mathematics, Repunit
• Index entries for linear recurrences with constant coefficients, signature (12,-11).

Crossrefs

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

Cf. A001020, A004191.

Programs

Magma

[(11^(n+1)-1)/10: n in [0..30]]; // G. C. Greubel, Feb 21 2024

Mathematica

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

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

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

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

Extensions

Title edited by Daniel Forgues, Jul 08 2011