A007925 a(n) = n^(n+1) - (n+1)^n.

-1, -1, -1, 17, 399, 7849, 162287, 3667649, 91171007, 2486784401, 74062575399, 2395420006033, 83695120256591, 3143661612445145, 126375169532421599, 5415486851106043649, 246486713303685957375, 11877172892329028459041, 604107995057426434824791

Offset: 0

Author

Dennis S. Kluk (mathemagician(AT)ameritech.net)

Keywords

• sign
• easy
• nice

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)

Examples

			a(2) = 1^2 - 2^1 = -1,
a(4) = 3^4 - 4^3 = 17.
		

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

Crossrefs

Cf. A051442.

Cf. A166326, A099498, A141074, A174379, A123206, A045575, A082754.

Programs

Maple

A007925:=n->n^(n+1)-(n+1)^n: seq(A007925(n), n=0..25); # Wesley Ivan Hurt, Jan 10 2017

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 *)

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

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.

a(n) = A007778(n) - A000169(n+1).

(End)

E.g.f.: LambertW(-x)/((1+LambertW(-x))*x)-LambertW(-x)/(1+LambertW(-x))^3. - Alois P. Heinz, Jul 04 2022