A002021 Pile of coconuts problem: (n-1)*(n^n - 1), n even; n^n - n + 1, n odd.

1, 3, 25, 765, 3121, 233275, 823537, 117440505, 387420481, 89999999991, 285311670601, 98077104930805, 302875106592241, 144456088732254195, 437893890380859361, 276701161105643274225, 827240261886336764161, 668888937280041138782191, 1978419655660313589123961

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

This is a generalization (from n = 5) of Ben Ames Williams's published problem. For a given n, the problem is effectively as follows. A successful monkey-share process removes 1 coconut for a monkey followed by an exact share of 1/n from the coconut pile. Determine the least initial number of coconuts for a monkey-share to succeed n times, leaving a multiple of n to be shared equally at the end. The problem in the D'Agostino link is slightly different, requiring a coconut for the monkey in the final division. - Peter Munn, Jun 14 2023

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• Anonymous, The Monkey and the Coconuts (with FormulaOne program)
• Santo D'Agostino, "The Coconut Problem"; Updated With Solution, May 2011.
• Mark Richardson, A Needlessly Complicated and Unhelpful Solution to Ben Ames Williams' Famous Coconuts Problem, The Winnower, Authorea (2016) Vol. 3.
• R. S. Underwood and Robert E. Moritz, Problem 3242, Amer. Math. Monthly, 35 (1928), 47-48.
• Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Crossrefs

Cf. A002022, A006091.

Programs

Maple

seq(`if`(n::even, (n-1)*(n^n - 1),n^n-n+1),n=1..30); # Robert Israel, Aug 26 2016

Mathematica

Table[If[EvenQ[n],(n-1)(n^n-1),n^n-n+1],{n,30}] (* Harvey P. Dale, Apr 21 2012 *)

Python

def a(n): return (n-1)*(n**n - 1) if n%2 == 0 else n**n - n + 1
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Feb 07 2022

Formula

E.g.f.: (1-x)*exp(x)-(W(x)+2)*(2*W(x)+1)/(2*(1+W(x))^3)-W(-x)/(2*(1+W(-x))^3) where W is the Lambert W function. - Robert Israel, Aug 26 2016

a(n) = 1-n-(-n)^n+(1+(-1)^n)*n^(n+1)/2. - Wesley Ivan Hurt, Nov 09 2023

Extensions

More terms from Harvey P. Dale, Apr 21 2012