OFFSET
0,5
COMMENTS
Conjecture: a(n) = floor((n - 3/2)/24^(1/4)) for n not in {0, 1, 6, 17, 2403, 5318}. - Charles R Greathouse IV, May 01 2012
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, pp. 55-57, Copernicus Press, NY, 1996.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number
FORMULA
a(n) = floor((A000332(n+3))^(1/4)) = floor(Ptop(n)^(1/4)) = floor(C(n+3, 4)^1/4) = floor((n * (n+1) * (n+2) * (n+3)/4!)^(1/4)).
a(n) = 0.4518... * n + O(1). - Charles R Greathouse IV, Dec 14 2015
EXAMPLE
a(3) = 1 because floor((3*4*5*6/24)^(1/4)) = floor(15^(1/4)) = floor(1.96798967) = 1.
MAPLE
a:= n-> floor(binomial(n+3, 4)^(1/4)):
seq(a(n), n=0..70); # Alois P. Heinz, Dec 14 2015
PROG
(PARI) a(n)=binomial(n+3, 4)^(1/4)\1 \\ Charles R Greathouse IV, May 01 2012
(PARI) a(n)=sqrtnint(binomial(n+3, 4), 4) \\ Charles R Greathouse IV, Dec 14 2015
(Magma) [Floor(Binomial(n+3, 4)^(1/4)): n in [3..70]]; // Vincenzo Librandi, Dec 14 2015
(Python)
from math import comb
from sympy import integer_nthroot
def A100682(n): return integer_nthroot(comb(n+3, 4), 4)[0] # Chai Wah Wu, Oct 02 2024
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Jonathan Vos Post, Dec 06 2004
STATUS
approved