0,3

Figurate numbers based on the 5-dimensional regular convex polytope, variously called the 5-dimensional hyperoctahedron, or the 5-dimensional cross-polytope, which is represented by the Schlaefli symbol {3, 3, 3, 4}. It is the dual of the 5-dimensional hypercube. Hyun Kwang Kim asserts that every nonnegative integer can be represented by the sum of no more than 14 of these 5-crosspolytope numbers. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 16 2004

If Y_i (i=1,2,3,4) are 2-blocks of a (n+4)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan R. Janjic (agnus(AT)blic.net), Oct 28 2007

H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.

Milan Janjic, Two Enumerative Functions

H. K. Kim, "On Regular polytope numbers".

J. V. Post, Table of polytope numbers, Sorted, Through 1,000,000.

J. V. Post, Math Pages.

Recurrence: a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6).

a(n) = 5-crosspolytope(n) = n*(2*n^4 + 10*n^2 + 3)/15. E.g. a(5) = 501 because 5*(2*5^4 + 10*5^2 + 3)/15 = 501. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 16 2004

a(n) = C(n+4,5) + 4 C(n+3,5) + 6 C(n+2,5) + 4 C(n+1,5) + C(n,5)

al:=proc(s, n) binomial(n+s-1, s); end; be:=proc(d, n) local r; add( (-1)^r*binomial(d-1, r)*2^(d-1-r)*al(d-r, n), r=0..d-1); end; [seq(be(5, n), n=0..100)];

Cf. A000332, A014820, A005900.

Adjacent sequences: A069035 A069036 A069037 this_sequence A069039 A069040 A069041

Sequence in context: A106041 A124162 A077044 this_sequence A030183 A135242 A041186

nonn

Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 03 2002