A085283 a(n) = n*n^n - (n-1)*(n-1)^n.

1, 1, 7, 65, 781, 11529, 201811, 4085185, 93864121, 2413042577, 68618940391, 2138428376721, 72470493235141, 2653457921150425, 104382202543721467, 4390455017903519489, 196621779843659466481, 9340717969198079777313

Offset: 0

Author

Paul Barry, Jun 26 2003

Keywords

• easy
• nonn

Comments

The system of equations

x(0) = n*x(1) + 1,

(n-1)*x(1) = n*x(2) + 1,

...

(n-1)*x(n) = n*x(n+1) + 1.

relates to the Monkey-And-Coconuts problem and reduces to the single equation

A007778(n-1)*x(0) = A007778(n)*x(n+1) + a(n),

whose solutions {x(0),x(n+1)} are given by {A014293(n), A085606(n)=A007778(n-1) - 1}. - Lekraj Beedassy, Jul 15 2003

For n >= 1, a(n) is equal to the number of functions f: {1,2,...,n+1}->{1,2,...,n} such that Im(f) contains a fixed element. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Links

• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• BBC, The Monkey and the Coconuts
• K. Belcourt, How Many Coconuts
• Santo D'Agostino, "The Coconut Problem"; Updated With Solution, May 2011.
• J. Dean, Sailors, monkey and coconuts
• A. K. Dewdney, The Monkey and the Coconuts
• G. Lewandrowski, The Monkey Problem
• R. Raja, Monkeys and Coconuts
• A. Rishi, Nemo's sailors and the monkey
• Dr. Rob, The MathForum, Coconut piles
• D. J. Wright, Five Pirates and a Monkey

Crossrefs

Cf. A045531, A055869.

Programs

Mathematica

Join[{1},Table[n*n^n-(n-1)(n-1)^n,{n,20}]] (* Harvey P. Dale, Sep 08 2016 *)

Formula

E.g.f.: -(x + 2*x*W(-x) + W(-x)^2)/(W(-x)*(1 + W(-x))^3), where W(x) is the Lambert W function. - Fabian Pereyra, Sep 26 2023