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 A100157 Structured rhombic dodecahedral numbers (vertex structure 9). 21
 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A005915 = alternate vertex; A100145 for more on structured polyhedral numbers. Sequence in context: A114012 A140784 A022285 * A144555 A192846 A212347 Adjacent sequences:  A100154 A100155 A100156 * A100158 A100159 A100160 KEYWORD easy,nonn AUTHOR James A. Record (james.record(AT)gmail.com), Nov 07 2004 STATUS approved

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Last modified January 22 03:32 EST 2021. Contains 340360 sequences. (Running on oeis4.)