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

0,3

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

Table of n, a(n) for n=0..17.

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

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

Mathematica

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

Crossrefs

Cf. A045531, A055869.

Sequence in context: A371393 A355163 A300488 * A069015 A097819 A152525

Adjacent sequences: A085280 A085281 A085282 * A085284 A085285 A085286

Keyword

easy,nonn

Author

Paul Barry, Jun 26 2003

Status

approved