A351331 a(n) = (n+1)*(n^n) - n + 1.

2, 11, 106, 1277, 18746, 326587, 6588338, 150994937, 3874204882, 109999999991, 3423740047322, 115909305827317, 4240251492291530, 166680102383370227, 7006302246093749986, 313594649253062377457, 14890324713954061755170, 747581753430634213933039, 39568393113206271782479562

Offset: 1

Author

Adam Thornton, Feb 07 2022

Keywords

• nonn
• easy

Comments

Arises in studying the "Pile of pairs of coconuts (and pals)" problem.

Motivated by a question passed along by Timothy Hunt from Kara Goeke; it is a generalization of the exercise in the first reference, which asks for a solution for n=3. For odd-indexed terms, the index may be taken as the number of participants in the circle. (The game doesn't work for even numbers of participants.)

In the case in the exercise: there are three participants splitting a box of chocolates whose number is even. The first person takes one, notes that the remainder is divisible by three, takes 1/3 of the remaining chocolates, passes the remainder to the second person, who takes one, notes that the remainder is divisible by three, takes 1/3 of the remaining chocolates, and passes the remainder to the last person, who takes one, notes that the remainder is divisible by three, and takes 1/3 of the remainder. The number of chocolates remaining after this last division are once again divisible by three.

The generalization that this sequence solves is that there are any odd number of people n in the circle, and that they each take 1/n of the remainder after taking their initial single item; the final number left is divisible by n as well.

The "Pals" part is that although this only represents solutions to the problem for an odd number of participants, the formula that generates those solutions is perfectly well-behaved for even n as well, and those may as well be terms in the sequence.

This is the same basic problem as A002021, with the further constraint that the initial number of coconuts be even.

References

• Mark Dugopolski, College Algebra, Addison-Wesley, 1995, page 16, exercise 123.

Crossrefs

Cf. A002021.

Programs

Maple

seq((n+1)*(n^n)-n+1, n=1..19); # Georg Fischer, Apr 10 2024

Extensions

Definition corrected by Georg Fischer, Apr 10 2024