0,3

Figurate numbers based on the 6-dimensional regular convex polytope called the 6-dimensional cross-polytope, or 6-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 4}. It is the dual of the 6-dimensional hypercube. Kim asserts that every nonnegative integer can be represented by the sum of no more than 19 of these 6-crosspolytope numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 16 2004

Starting with "1" = binomial transform of [1, 11, 50, 120, 160, 112, 32, 0, 0, 0,...] where (1, 11, 50, 120, 160, 112, 32) = row 6 of the Chebyshev triangle A081277. Also = row 6 of the array in A142978. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008

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

J. V. Post, "4-Dimensional Jonathan numbers: polytope numbers and Centered polytope numbers of Higher Than 3 Dimensions", Draft 1.5 of 9 a.m., 12 March 2004, circulated by e-mail.

Index entries for sequences related to linear recurrences with constant coefficients

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) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).

a(n) = 6-crosspolytope(n) = (n^2)*(2*n^4 + 20*n^2 + 23 )/45. E.g. a(12) = 142000 because (12^2)*(2*12^4 + 20*12^2 + 23 )/45. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 16 2004

Contribution from Stephen Crowley (crow(AT)crowlogic.net), Jul 14 2009: (Start)

sum(1/((1/45)*n^2*(2*n^4+20*n^2+23)),n=1..infinity)=-5*(Sum(_alpha*(77*_alpha^2+655)*Psi(1-_alpha), _alpha = RootOf(2*_Z^4+20*_Z^2+23)))*(1/3174)+15*Pi^2*(1/46)=1.10203455013915915542552577192042916250524...

sum(1/(((1/45)*n^2*(2*n^4+20*n^2+23)*n!)),n=1..infinity)=hypergeom([1, 1, 1, 1-a, 1+b, 1-b, 1+a], [2, 2, 2, 2+b, 2-b, 2+a, 2-a], 1)=1.04409584723862654376639417281585634150689... where a=I*sqrt(20+6*sqrt(6))*(1/2) and b=I*sqrt(20-6*sqrt(6))*(1/2) (End)

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(6, n), n=0..100)];

Cf. A000332, A014820, A005900, A069038, A099193, A099195.

Cf. A081277, A142978.

Sequence in context: A120783 A103475 A024014 this_sequence A156196 A041270 A055912

Adjacent sequences: A069036 A069037 A069038 this_sequence A069040 A069041 A069042

nonn

Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 03 2002