A100157 Structured rhombic dodecahedral numbers (vertex structure 9).

1, 14, 55, 140, 285, 506, 819, 1240, 1785, 2470, 3311, 4324, 5525, 6930, 8555, 10416, 12529, 14910, 17575, 20540, 23821, 27434, 31395, 35720, 40425, 45526, 51039, 56980, 63365, 70210, 77531, 85344, 93665, 102510, 111895, 121836, 132349, 143450, 155155, 167480

Offset: 1

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

• easy
• nonn

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

Examples

			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
		

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).

Crossrefs

Cf. A005915 = alternate vertex; A100145 for more on structured polyhedral numbers.

Cf. A000330, A000326, A006918, A008610, A053307, A100171, A100185, A213752, A234043.

Programs

Magma

[(1/6)*(16*n^3-12*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

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

PARI

a(n)=(16*n^3-12*n^2+2*n)/6 \\ Charles R Greathouse IV, Sep 24 2015

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(2*n+1,3) + binomial(2*n,3). - John Molokach, Jul 10 2013

a(n) = Sum_{i=-(n-1)..(n-1)} (n+i)^2. - Bruno Berselli, Jul 24 2014

From Elmo R. Oliveira, Aug 04 2025: (Start)

E.g.f.: exp(x)*x*(8*x^2 + 18*x + 3)/3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)