A005095 a(n) = n! + n.

1, 2, 4, 9, 28, 125, 726, 5047, 40328, 362889, 3628810, 39916811, 479001612, 6227020813, 87178291214, 1307674368015, 20922789888016, 355687428096017, 6402373705728018, 121645100408832019, 2432902008176640020, 51090942171709440021

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Every infinite, increasing, integer arithmetic progression meets this sequence infinitely often. - John Abbott (abbott(AT)dima.unige.it), Mar 06 2003

Sum(A010051(k): A038507(n) < k <= a(n)) = 0. - Reinhard Zumkeller, Jul 10 2009

Largest k such that (k!-n!)/(k-n) is an integer. - Derek Orr, Apr 02 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 5.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A135723.

Cf. A090786. - Reinhard Zumkeller, Jul 10 2009

Programs

Magma

[Factorial(n) + n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2013

Mathematica

lst={};Do[AppendTo[lst, n!+n], {n, 3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[n! + n, {n, 0, 20}] (* Vincenzo Librandi, Jun 08 2013 *)

Maxima

A005095(n):= n!+n$
makelist(A005095(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

Formula

E.g.f.: x*exp(x) + 1/(1-x). - Len Smiley, Dec 05 2001

Row sums of triangle A135723. - Gary W. Adamson, Nov 25 2007

(n-1)*(n-3)*a(n) -n*(n^2-3*n+1)*a(n-1) +n*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Oct 30 2015

a(n) +(-n-3)*a(n-1) +3*(n)*a(n-2) +(-3*n+5)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Oct 30 2015