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%I #58 Mar 14 2023 09:30:47
%S 1,13,157,1885,22621,271453,3257437,39089245,469070941,5628851293,
%T 67546215517,810554586205,9726655034461,116719860413533,
%U 1400638324962397,16807659899548765,201691918794585181
%N Expansion of 1/((1-x)*(1-12*x)).
%C 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
%C 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
%C Numbers that are repunits in duodecimal representation. - _Reinhard Zumkeller_, Dec 12 2012
%C 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
%H Vincenzo Librandi, <a href="/A016125/b016125.txt">Table of n, a(n) for n = 0..300</a>
%H Kival Ngaokrajang, <a href="/A016125/a016125.pdf">Illustration of initial terms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Duodecimal.html">Duodecimal</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Duodecimal">Duodecimal</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Repunit">Repunit</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-12).
%F a(n) = (12^(n+1) - 1)/11.
%F a(n) = 12*a(n-1)+1 for n>0, a(0)=1. - _Vincenzo Librandi_, Nov 19 2010
%F a(n) = Sum_{i=0...n} 11^i*binomial(n+1,n-i). - _Bruno Berselli_, Nov 11 2015
%F E.g.f.: exp(x)*(12*exp(11*x) - 1)/11. - _Stefano Spezia_, Mar 11 2023
%e For n=5, a(5) = 1*6 + 11*15 + 121*20 + 1331*15 + 14641*6 + 161051*1 = 271453. - _Bruno Berselli_, Nov 11 2015
%p a:=n->sum(12^(n-j),j=1..n): seq(a(n), n=1..17); # _Zerinvary Lajos_, Jan 04 2007
%t Join[{a=1,b=13},Table[c=13*b-12*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2011 *)
%t CoefficientList[Series[1/((1-x)(1-12x)),{x,0,20}],x] (* or *) LinearRecurrence[{13,-12},{1,13},20] (* _Harvey P. Dale_, Aug 20 2022 *)
%o (Sage) [lucas_number1(n,13,12) for n in range(1,18)] # _Zerinvary Lajos_, Apr 29 2009
%o (Sage) [gaussian_binomial(n,1,12) for n in range(1,18)] # _Zerinvary Lajos_, May 28 2009
%o (Sage) [(12^(n+1)-1)/11 for n in (0..20)] # _Bruno Berselli_, Nov 11 2015
%o (Magma) [(12^(n+1)-1)/11: n in [0..20]]; // _Vincenzo Librandi_, Jul 01 2011
%o (PARI) Vec(1/(1-13*x+12*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jul 01 2011
%o (Maxima) A016125(n):=(12^(n+1) - 1)/11$
%o makelist(A016125(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Haskell)
%o a016125 n = a016125_list !! n
%o a016125_list = iterate ((+ 1) . (* 12)) 1
%o -- _Reinhard Zumkeller_, Dec 12 2012
%Y Cf. A001021, A024140, A178248.
%Y Cf. A001020, A135278.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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