A007611 - OEIS

%I M0807 #55 Apr 09 2023 07:53:48

%S 2,3,6,14,40,152,784,5168,40576,363392,3629824,39918848,479005696,

%T 6227028992,87178307584,1307674400768,20922789953536,355687428227072,

%U 6402373705990144,121645100409356288,2432902008177688576

%N a(n) = n! + 2^n.

%C Answer to the problem A1 proposed during the 51th Putnam Competition in 1990 (see Kedlaya and Larson links). - _Bernard Schott_, Mar 29 2023

%D Ron Larson, Robert P. Hostetler and Bruce H. Edwards, Exercise 135 in Calculus of a Single Variable for Advanced High School Students, Page 605, 8th Edition, 2006.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A007611/b007611.txt">Table of n, a(n) for n = 0..200</a>.

%H Kiran S. Kedlaya, <a href="http://kskedlaya.org/putnam-archive/1990.pdf">Problem A-1</a>, The 51th William Lowell Putnam Mathematical Competition, Dec 08 1990.

%H Loren Larson, <a href="http://www.jstor.org/stable/2690769">Problem A-1</a>, 51st Annual Putnam Competition, Math. Mag., 64 (1991), 141-144.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical Competitions</a>.

%F a(n) = (n+4)*a(n-1) - 4*n*a(n-2) + (4*n-8)*a(n-3) with a(0) = 2, a(1) = 3, a(2) = 6. - Dennis S. Kluk (mathemagician(AT)ameritech.net), Feb 12 2006

%F From _Robert Israel_, Mar 06 2016: (Start)

%F a(n) = A000079(n) + A000142(n).

%F E.g.f.: 1/(1-x) + exp(2x). (End)

%p seq(2^n+n!, n=0..50); # _Robert Israel_, Mar 06 2016

%t Table[n! + 2^n, {n, 0, 19}] (* _Alonso del Arte_, Mar 06 2016 *)

%o (Sage) [factorial(n)+2^n for n in range(0, 21)] # _Zerinvary Lajos_, Oct 27 2009

%o (Magma) [Factorial(n)+ 2^n: n in [0..30]]; // _Vincenzo Librandi_, Jun 10 2013

%o (PARI) a(n) = n! + 2^n; \\ _Michel Marcus_, Mar 30 2023

%Y Cf. A000079, A000142.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_