A016125 Expansion of 1/((1-x)*(1-12*x)).

1, 13, 157, 1885, 22621, 271453, 3257437, 39089245, 469070941, 5628851293, 67546215517, 810554586205, 9726655034461, 116719860413533, 1400638324962397, 16807659899548765, 201691918794585181

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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>=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]:=13, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-2)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010

Numbers that are repunits in duodecimal representation. - Reinhard Zumkeller, Dec 12 2012

a(n) is the total number of holes in a certain box fractal (start with 12 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

Examples

			For n=5, a(5) = 1*6 + 11*15 + 121*20 + 1331*15 + 14641*6 + 161051*1 = 271453. - _Bruno Berselli_, Nov 11 2015
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Kival Ngaokrajang, Illustration of initial terms
• Eric Weisstein's World of Mathematics, Repunit
• Eric Weisstein's World of Mathematics, Duodecimal
• Wikipedia, Duodecimal
• Wikipedia, Repunit
• Index entries for linear recurrences with constant coefficients, signature (13,-12).

Crossrefs

Cf. A001021, A024140, A178248.

Cf. A001020, A135278.

Programs

Haskell

a016125 n = a016125_list !! n
a016125_list = iterate ((+ 1) . (* 12)) 1
-- Reinhard Zumkeller, Dec 12 2012

Magma

[(12^(n+1)-1)/11: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011

Maple

a:=n->sum(12^(n-j),j=1..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 04 2007

Mathematica

Join[{a=1,b=13},Table[c=13*b-12*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[1/((1-x)(1-12x)),{x,0,20}],x] (* or *) LinearRecurrence[{13,-12},{1,13},20] (* Harvey P. Dale, Aug 20 2022 *)

Maxima

A016125(n):=(12^(n+1) - 1)/11$
makelist(A016125(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

Vec(1/(1-13*x+12*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

Sage

[lucas_number1(n,13,12) for n in range(1,18)] # Zerinvary Lajos, Apr 29 2009

Sage

[gaussian_binomial(n,1,12) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

Sage

[(12^(n+1)-1)/11 for n in (0..20)] # Bruno Berselli, Nov 11 2015

Formula

a(n) = (12^(n+1) - 1)/11.

a(n) = 12*a(n-1)+1 for n>0, a(0)=1. - Vincenzo Librandi, Nov 19 2010

a(n) = Sum_{i=0...n} 11^i*binomial(n+1,n-i). - Bruno Berselli, Nov 11 2015

E.g.f.: exp(x)*(12*exp(11*x) - 1)/11. - Stefano Spezia, Mar 11 2023