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A100157
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Structured rhombic dodecahedral numbers (vertex structure 9).
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6
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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; internal format)
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OFFSET
| 1,2
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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 R. Janjic (agnus(AT)blic.net), 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). [From John M. Campbell, Sep 07 2011]
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REFERENCES
| Jolley, Summation of Series, Dover (1961).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Milan Janjic, Two Enumerative Functions
Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).
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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). [From Reinhard Zumkeller, Jul 06 2009]
sum_{n=1..infinity} 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
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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 (zerinvarylajos(AT)yahoo.com), Jan 02 2008
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PROG
| (MAGMA) [(1/6)*(16*n^3-12*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
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CROSSREFS
| Cf. A005915 = alternate vertex; A100145 for more on structured polyhedral numbers.
Sequence in context: A114012 A140784 A022285 * A144555 A192846 A115129
Adjacent sequences: A100154 A100155 A100156 * A100158 A100159 A100160
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KEYWORD
| easy,nonn
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AUTHOR
| James A. Record (james.record(AT)gmail.com). Nov 07, 2004.
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