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, 469162909670186351510479, 24841737241149880018918225

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..19.

BBC, The Monkey and the Coconuts.

Kenneth Belcourt, How Many Coconuts.

Santo D'Agostino, "The Coconut Problem"; Updated With Solution, May 2011.

Jimmie Dean, Sailors, monkey and coconuts.

A. K. Dewdney, The Monkey and the Coconuts.

Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.

Gary Lewandrowski, The Monkey Problem.

Ravi Raja, Monkeys and Coconuts.

Aditya Rishi, Nemo's sailors and the monkey.

Doctor Rob, Coconut piles, The Math Forum, 1998.

David J. Wright, Five Pirates and a Monkey. [Dead link]

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

a(n) ~ (1 - exp(-1-3/(2*n))) * n^(n+1). - Amiram Eldar, Feb 07 2026

Mathematica

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

Crossrefs

Cf. A007778, A045531, A055869.

Sequence in context: A355163 A390374 A300488 * A069015 A097819 A152525

Adjacent sequences: A085280 A085281 A085282 * A085284 A085285 A085286

Keyword

easy,nonn

Author

Paul Barry, Jun 26 2003

Status

approved