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A100157 Structured rhombic dodecahedral numbers (vertex structure 9). 21


%S 1,14,55,140,285,506,819,1240,1785,2470,3311,4324,5525,6930,8555,

%T 10416,12529,14910,17575,20540,23821,27434,31395,35720,40425,45526,

%U 51039,56980,63365,70210,77531,85344

%N Structured rhombic dodecahedral numbers (vertex structure 9).

%C Also structured triakis octahedral numbers (vertex structure 9) (Cf. A100171 = alternate vertex); and structured heptagonal anti-prism numbers (Cf. A100185 = structured anti-prisms).

%C 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

%C 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

%C Principal diagonal of the convolution array A213752. - _Clark Kimberling_, Jun 20 2012

%C 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

%D Jolley, Summation of Series, Dover (1961).

%H Vincenzo Librandi, <a href="/A100157/b100157.txt">Table of n, a(n) for n = 1..5000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H A. Schuetz and G. Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.

%H StackExchange, <a href="http://math.stackexchange.com/questions/29125/what-is-a-structured-polyhedron">What is a Structured Polyhedron?</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (16*n^3-12*n^2+2*n)/6.

%F a(n) = n*(2*n-1)*(4*n-1)/3 = A000330(2*n-1). - _Reinhard Zumkeller_, Jul 06 2009

%F sum_{n>=1} 1/(24*a(n)) = Pi/8-log(2)/2 = 0.046125491418751.. [Jolley eq. 251]

%F G.f. x*(1+10*x+5*x^2) / (x-1)^4 . - _R. J. Mathar_, Oct 03 2011

%F a(n) = binomial(2n+1,3) + binomial(2n,3). - _John Molokach_, Jul 10 2013

%F a(n) = sum( (n+i)^2, i=-(n-1)..(n-1) ). - _Bruno Berselli_, Jul 24 2014

%e 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

%p 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

%o (MAGMA) [(1/6)*(16*n^3-12*n^2+2*n): n in [1..40]]; // _Vincenzo Librandi_, Jul 19 2011

%o (PARI) a(n)=(16*n^3-12*n^2+2*n)/6 \\ _Charles R Greathouse IV_, Sep 24 2015

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

%K easy,nonn

%O 1,2

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

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Last modified March 3 12:16 EST 2021. Contains 341762 sequences. (Running on oeis4.)