|
%I #30 Sep 08 2022 08:45:15
%S 0,1,1,1,2,2,3,3,4,4,5,5,6,6,6,7,7,8,8,9,9,10,10,11,11,11,12,12,13,13,
%T 14,14,15,15,16,16,16,17,17,18,18,19,19,20,20,21,21,21,22,22,23,23,24,
%U 24,25,25,25,26,26,27,27,28,28,29,29,30,30,30,31,31
%N Floor of 4th root of pentatope numbers.
%C 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
%D J. H. Conway and R. K. Guy, The Book of Numbers, pp. 55-57, Copernicus Press, NY, 1996.
%H Alois P. Heinz, <a href="/A100682/b100682.txt">Table of n, a(n) for n = 0..10000</a>
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2003), 65-75.
%H Jonathan Vos Post, <a href="http://magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentatopeNumber.html">Pentatope Number</a>
%F 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))
%F a(n) = 0.4518... * n + O(1). - _Charles R Greathouse IV_, Dec 14 2015
%e a(3) = 1 because floor((3*4*5*6/24)^(1/4)) = floor(15^(1/4)) = floor(1.96798967)) = 1.
%p a:= n-> floor(binomial(n+3, 4)^(1/4)):
%p seq(a(n), n=0..70); # _Alois P. Heinz_, Dec 14 2015
%o (PARI) a(n)=binomial(n+3,4)^(1/4)\1 \\ _Charles R Greathouse IV_, May 01 2012
%o (PARI) a(n)=sqrtnint(binomial(n+3,4),4) \\ _Charles R Greathouse IV_, Dec 14 2015
%o (Magma) [Floor(Binomial(n+3, 4)^(1/4)): n in [3..70]]; // _Vincenzo Librandi_, Dec 14 2015
%Y Cf. A000332, A100009, A007501, A099179.
%K easy,nonn
%O 0,5
%A _Jonathan Vos Post_, Dec 06 2004
|
