OFFSET
0,4
COMMENTS
From Mathew Englander, Jul 07 2020: (Start)
All a(n) are odd and for n even, a(n) == 3 (mod 4); for n odd and n != 1, a(n) == 1 (mod 4).
The correspondence between n and a(n) when considered mod 6 is as follows: for n == 0, 1, 2, or 3, a(n) == 5; for n == 4, a(n) == 3; for n == 5, a(n) == 1.
For all n, a(n)+1 is a multiple of n^2.
For n odd and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs of the above, see the Englander link. (End)
REFERENCES
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
Mathew Englander, Notes on OEIS A007925
Sergio Silva, Teste Numerico, Item 3.
H. J. Smith, Contour Plot of z = x^y - y^x
FORMULA
Asymptotic expression for a(n) is a(n) ~ n^n * (n - e). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
From Mathew Englander, Jul 07 2020: (Start)
a(n) = A111454(n+4) - 1.
a(n) = A055651(n, n+1).
a(n) = A220417(n+1, n) for n >= 1.
(End)
E.g.f.: LambertW(-x)/((1+LambertW(-x))*x)-LambertW(-x)/(1+LambertW(-x))^3. - Alois P. Heinz, Jul 04 2022
EXAMPLE
a(2) = 1^2 - 2^1 = -1,
a(4) = 3^4 - 4^3 = 17.
MAPLE
MATHEMATICA
lst={}; Do[AppendTo[lst, (n^(n+1)-((n+1)^n))], {n, 0, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
#^(#+1)-(#+1)^#&/@Range[0, 20] (* Harvey P. Dale, Oct 22 2011 *)
PROG
(Maxima) A007925[n]:=n^(n+1)-(n+1)^n$ makelist(A007925[n], n, 0, 30); /* Martin Ettl, Oct 29 2012 */
(PARI) a(n)=n^(n+1)-(n+1)^n \\ Charles R Greathouse IV, Feb 06 2017
KEYWORD
sign,easy,nice
AUTHOR
Dennis S. Kluk (mathemagician(AT)ameritech.net)
STATUS
approved