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%I #47 Jun 29 2023 18:38:11
%S 1,18,307,5220,88741,1508598,25646167,435984840,7411742281,
%T 125999618778,2141993519227,36413889826860,619036127056621,
%U 10523614159962558,178901440719363487,3041324492229179280,51702516367896047761
%N Partial sums of powers of 17 (A001026).
%C 17^a(n) is largest power of 17 dividing (17^n)!.
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=17, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - _Milan Janjic_, Feb 21 2010
%H Robert Israel, <a href="/A091045/b091045.txt">Table of n, a(n) for n = 1..812</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18, -17).
%F a(n) = Sum_{k=0..n-1} 17^k = (17^n - 1)/16.
%F G.f.: x/((1 - 17*x)*(1 - x))= (1/(1 - 17*x) - 1/(1 - x))/16.
%F a(n) = 17*a(n-1)+1 (with a(1)=1). - _Vincenzo Librandi_, Nov 16 2010
%F E.g.f.: exp(9*x)*sinh(8*x)/8. - _Stefano Spezia_, Mar 11 2023
%p ListTools:-PartialSums([seq(17^k,k=0..30)]); # _Robert Israel_, Feb 18 2018
%t Table[17^n, {n, 0, 16}] // Accumulate (* _Jean-François Alcover_, Jul 05 2013 *)
%o (Sage) [gaussian_binomial(n,1,17) for n in range(1,18)] # _Zerinvary Lajos_, May 28 2009
%o (Maxima) makelist(sum(17^k, k, 0, n), n, 0, 30); /* _Martin Ettl_, Nov 05 2012 */
%o (Magma) [&+[17^i: i in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Feb 19 2018
%Y Cf. similar sequences of the form (k^n-1)/(k-1) with k prime: A000225 (k=2), A003462 (k=3), A003463 (k=5), A023000 (k=7), A016123 (k=11), A091030 (k=13), this sequence (k=17), A218722 (k=19), A218726 (k=23), A218732 (k=29), A218734 (k=31), A218740 (k=37), A218744 (k=41), A218746 (k=43), A218750 (k=47).
%Y Cf. A001026.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Jan 23 2004
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