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

0,2

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

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).

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

Example

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

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 *)

Prog

(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
(Magma) [(12^(n+1)-1)/11: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011
(PARI) Vec(1/(1-13*x+12*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011
(Maxima) A016125(n):=(12^(n+1) - 1)/11$
makelist(A016125(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(Haskell)
a016125 n = a016125_list !! n
a016125_list = iterate ((+ 1) . (* 12)) 1
-- Reinhard Zumkeller, Dec 12 2012

Crossrefs

Cf. A001021, A024140, A178248.

Cf. A001020, A135278.

Sequence in context: A159499 A125470 A165151 * A353147 A175519 A015470

Adjacent sequences: A016122 A016123 A016124 * A016126 A016127 A016128

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved