OFFSET
1,2
COMMENTS
Also structured triakis octahedral numbers (vertex structure 9) (Cf. A100171 = alternate vertex); and structured heptagonal anti-prism numbers (Cf. A100185 = structured anti-prisms).
If Y is a 2-subset of a 2n-set X then, for n>=2, a(n-1) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Let M(2n-1) be a (2n-1)x(2n-1) matrix whose (i,j)-entry equals i^2/(i^2+sqrt(-1)) if i=j and equals 1 otherwise. Then a(n) equals (-1)^(n+1) times the real part of prod(k^2+sqrt(-1),k=1...2n-1) times the determinant of M(2n-1). - John M. Campbell, Sep 07 2011
Principal diagonal of the convolution array A213752. - Clark Kimberling, Jun 20 2012
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n)= Cat(n,4), so enumerates the number of (n+1)-gon partitions of a (4*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A234043 (k=5). Also, a(n)= A006918(4n+1) = A008610(4n+1) = A053307(4n+1) with offset=0. - Tom Copeland, Oct 05 2014
REFERENCES
Jolley, Summation of Series, Dover (1961).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Milan Janjic, Two Enumerative Functions
A. Schuetz and G. Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
StackExchange, What is a Structured Polyhedron?
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = (16*n^3-12*n^2+2*n)/6.
a(n) = n*(2*n-1)*(4*n-1)/3 = A000330(2*n-1). - Reinhard Zumkeller, Jul 06 2009
sum_{n>=1} 1/(24*a(n)) = Pi/8-log(2)/2 = 0.046125491418751.. [Jolley eq. 251]
G.f. x*(1+10*x+5*x^2) / (x-1)^4 . - R. J. Mathar, Oct 03 2011
a(n) = binomial(2n+1,3) + binomial(2n,3). - John Molokach, Jul 10 2013
a(n) = sum( (n+i)^2, i=-(n-1)..(n-1) ). - Bruno Berselli, Jul 24 2014
EXAMPLE
For n=4, sum( (4+i)^2, i=-3..3 ) = (4-3)^2+(4-2)^2+(4-1)^2+(4-0)^2+(4+1)^2+(4+2)^2+(4+3)^2 = 140 = a(4). - Bruno Berselli, Jul 24 2014
MAPLE
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m*4), m=1..32) ; # Zerinvary Lajos, Jan 02 2008
PROG
(Magma) [(1/6)*(16*n^3-12*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
(PARI) a(n)=(16*n^3-12*n^2+2*n)/6 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
James A. Record (james.record(AT)gmail.com), Nov 07 2004
STATUS
approved