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 A016123 a(n) = (11^(n+1) - 1)/10. 53
 1, 12, 133, 1464, 16105, 177156, 1948717, 21435888, 235794769, 2593742460, 28531167061, 313842837672, 3452271214393, 37974983358324, 417724816941565, 4594972986357216, 50544702849929377, 555991731349223148 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 11^a(n) is highest power of 11 dividing (11^(n+1))!. Partial sums of powers of 11 (A001020). a(n)=[(11^n)-1]/10 - Ctibor O. Zizka, Feb 18 2008 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 Eric Weisstein's World of Mathematics, Repunit Index entries for linear recurrences with constant coefficients, signature (12,-11). FORMULA a(n)= sum(11^k, k=0..n) = (11^(n+1)-1)/10. G.f.: (1/(1-11*x)-1/(1-x))/(10*x)=1/((1-11*x)*(1-x)). For analogs with primes 2, 3, 5, 7, 13 and 17 see: A000225, A003462, A003463, A023000, A091030 and A091045, respectively. a(0)=1, a(n)=11*a(n-1)+1. [From Vincenzo Librandi, Feb 05 2011] a(0)=0, a(1)=1, a(n)=12*a(n-1)-11*a(n-2) [From Harvey P. Dale, Apr 05 2012] MAPLE a:=n->sum(11^(n-j), j=1..n): seq(a(n), n=1..18); - Zerinvary Lajos, Jan 04 2007 MATHEMATICA Join[{a=1, b=12}, Table[c=12*b-11*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011*) (11^Range[0, 20]-1)/10 (* or *) LinearRecurrence[{12, -11}, {0, 1}, 20] (* Harvey P. Dale, Apr 05 2012 *) PROG (Sage) [lucas_number1(n, 12, 11) for n in xrange(1, 19)]# - Zerinvary Lajos, Apr 27 2009 (Sage) [gaussian_binomial(n, 1, 11) for n in xrange(1, 19)] # - Zerinvary Lajos, May 28 2009 (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 CROSSREFS Cf. A004191. Sequence in context: A120673 A120674 A244205 * A015457 A015469 A144785 Adjacent sequences:  A016120 A016121 A016122 * A016124 A016125 A016126 KEYWORD nonn,easy,changed AUTHOR EXTENSIONS Title edited by Daniel Forgues, Jul 08 2011 STATUS approved

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