A007611 a(n) = n! + 2^n.

2, 3, 6, 14, 40, 152, 784, 5168, 40576, 363392, 3629824, 39918848, 479005696, 6227028992, 87178307584, 1307674400768, 20922789953536, 355687428227072, 6402373705990144, 121645100409356288, 2432902008177688576

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• easy

Comments

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200.
• Kiran S. Kedlaya, Problem A-1, The 51th William Lowell Putnam Mathematical Competition, Dec 08 1990.
• Loren Larson, Problem A-1, 51st Annual Putnam Competition, Math. Mag., 64 (1991), 141-144.
• Index entries for sequences related to factorial numbers.
• Index to sequences related to Olympiads and other Mathematical Competitions.

Crossrefs

Cf. A000079, A000142.

Programs

Magma

[Factorial(n)+ 2^n: n in [0..30]]; // Vincenzo Librandi, Jun 10 2013

Maple

seq(2^n+n!, n=0..50); # Robert Israel, Mar 06 2016

Mathematica

Table[n! + 2^n, {n, 0, 19}] (* Alonso del Arte, Mar 06 2016 *)

PARI

a(n) = n! + 2^n; \\ Michel Marcus, Mar 30 2023

Sage

[factorial(n)+2^n for n in range(0, 21)] # Zerinvary Lajos, Oct 27 2009

Formula

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

From Robert Israel, Mar 06 2016: (Start)

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

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