A091045 Partial sums of powers of 17 (A001026).

1, 18, 307, 5220, 88741, 1508598, 25646167, 435984840, 7411742281, 125999618778, 2141993519227, 36413889826860, 619036127056621, 10523614159962558, 178901440719363487, 3041324492229179280, 51702516367896047761

Offset: 1

Author

Wolfdieter Lang, Jan 23 2004

Keywords

• nonn
• easy

Comments

17^a(n) is largest power of 17 dividing (17^n)!.

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

Links

• Robert Israel, Table of n, a(n) for n = 1..812
• Index entries for linear recurrences with constant coefficients, signature (18, -17).

Crossrefs

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).

Cf. A001026.

Programs

Magma

[&+[17^i: i in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Feb 19 2018

Maple

ListTools:-PartialSums([seq(17^k,k=0..30)]); # Robert Israel, Feb 18 2018

Mathematica

Table[17^n, {n, 0, 16}] // Accumulate (* Jean-François Alcover, Jul 05 2013 *)

Maxima

makelist(sum(17^k, k, 0, n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

Sage

[gaussian_binomial(n,1,17) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

Formula

a(n) = Sum_{k=0..n-1} 17^k = (17^n - 1)/16.

G.f.: x/((1 - 17*x)*(1 - x))= (1/(1 - 17*x) - 1/(1 - x))/16.

a(n) = 17*a(n-1)+1 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010

E.g.f.: exp(9*x)*sinh(8*x)/8. - Stefano Spezia, Mar 11 2023