A037205 a(n) = (n+1)^n - 1.

0, 1, 8, 63, 624, 7775, 117648, 2097151, 43046720, 999999999, 25937424600, 743008370687, 23298085122480, 793714773254143, 29192926025390624, 1152921504606846975, 48661191875666868480, 2185911559738696531967, 104127350297911241532840, 5242879999999999999999999, 278218429446951548637196400, 15519448971100888972574851071

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n >= 1, a(n) = order of Fibonacci group F(n+1,n).

The terms, written in base n+1, are n digits of value n. For example, a(4) = 624 = 4444 in base 5. - Marc Morgenegg, Nov 30 2016

For n >= 1, in a square grid of side n, this is the number of ways to populate the grid with 1 X 1 blocks (with at least one block) so that no block falls under the effect of gravity. - Paolo Xausa, Apr 12 2021

For n > 1, (n-1)^2 | a(n). - David A. Corneth, Dec 15 2022

References

• D. L. Johnson, Presentation of Groups, Cambridge, 1976, p. 182.
• Richard M. Thomas, The Fibonacci groups revisited, in Groups - St. Andrews 1989, Vol. 2, 445-454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385
• Michael Penn, A divisibility problem., YouTube video, 2021.

Crossrefs

A diagonal of A202624.

Programs

Magma

[(n + 1)^n - 1: n in [0..25]]; // G. C. Greubel, Nov 10 2017

Mathematica

Table[(n + 1)^n - 1, {n, 0, 21}] (* or *)
Table[If[n < 1, Length@ #, FromDigits[#, n + 1]] &@ ConstantArray[n, n], {n, 0, 21}] (* Michael De Vlieger, Nov 30 2016 *)

PARI

for(n=0,25, print1((n + 1)^n - 1, ", ")) \\ G. C. Greubel, Nov 10 2017

Formula

a(n) = A000169(n+1) - 1 = A060072(n+1)*(n-1) = A060073(n+1)*(n-1)^2.

E.g.f.: 1/(exp(LambertW(-x)) - x) - exp(x). - Ilya Gutkovskiy, Nov 30 2016

E.g.f.: -exp(x) - 1/(x + x/LambertW(-x)). - Vaclav Kotesovec, Dec 05 2016

a(n) = Sum_{k=1..n} binomial(n,k)*n^k [from Paolo Xausa's comment]. - Joerg Arndt, Apr 12 2021

Extensions

Revised by N. J. A. Sloane, Dec 30 2011