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%I #49 Feb 21 2024 20:29:48
%S 1,12,133,1464,16105,177156,1948717,21435888,235794769,2593742460,
%T 28531167061,313842837672,3452271214393,37974983358324,
%U 417724816941565,4594972986357216,50544702849929377,555991731349223148
%N a(n) = (11^(n+1) - 1)/10.
%C 11^a(n) is highest power of 11 dividing (11^(n+1))!.
%C Partial sums of powers of 11 (A001020).
%C 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
%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>=2, a(n-1)=(-1)^n*charpoly(A,1). - _Milan Janjic_, Feb 21 2010
%H G. C. Greubel, <a href="/A016123/b016123.txt">Table of n, a(n) for n = 0..950</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-11).
%F a(n) = Sum_{k=0..n} 11^k = (11^(n+1) - 1)/10.
%F G.f.: (1/(1-11*x) - 1/(1-x))/(10*x) = 1/((1-11*x)*(1-x)).
%F a(0)=1, a(n) = 11*a(n-1) + 1. - _Vincenzo Librandi_, Feb 05 2011
%F a(0)=0, a(1)=1, a(n) = 12*a(n-1) - 11*a(n-2). - _Harvey P. Dale_, Apr 05 2012
%F E.g.f.: exp(x)*(11*exp(10*x) - 1)/10. - _Stefano Spezia_, Mar 11 2023
%t (11^Range[0,20]-1)/10 (* or *) LinearRecurrence[{12,-11},{0,1},20] (* _Harvey P. Dale_, Apr 05 2012 *)
%o (Sage) [lucas_number1(n,12,11) for n in range(1, 19)] # _Zerinvary Lajos_, Apr 27 2009
%o (Sage) [gaussian_binomial(n,1,11) for n in range(1,19)] # _Zerinvary Lajos_, May 28 2009
%o (Maxima) A016123(n):=(11^(n+1)-1)/10$
%o makelist(A016123(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (PARI) a(n)=(11^(n+1)-1)/10 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [(11^(n+1)-1)/10: n in [0..30]]; // _G. C. Greubel_, Feb 21 2024
%Y For analogs with primes 2, 3, 5, 7, 13 and 17 see: A000225, A003462, A003463, A023000, A091030 and A091045, respectively.
%Y Cf. A001020, A004191.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Title edited by _Daniel Forgues_, Jul 08 2011
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