A002021 Pile of coconuts problem: (n-1)*(n^n - 1), n even; n^n - n + 1, n odd. (Formerly M3114 N1262)

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

Offset

1,2

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

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

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 *)

Prog

(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

Crossrefs

Cf. A002022, A006091.

Sequence in context: A365359 A062411 A136516 * A322063 A350902 A306792

Adjacent sequences: A002018 A002019 A002020 * A002022 A002023 A002024

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Apr 21 2012

Status

approved