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%I #33 Sep 08 2022 08:45:14
%S 1,10,37,92,185,326,525,792,1137,1570,2101,2740,3497,4382,5405,6576,
%T 7905,9402,11077,12940,15001,17270,19757,22472,25425,28626,32085,
%U 35812,39817,44110,48701,53600,58817,64362,70245,76476,83065,90022,97357,105080,113201,121730
%N Cupolar numbers: a(n) = (n+1)*(5*n^2+7*n+3)/3.
%C Number of equal balls that will fill a triangular cupola, formed by splitting a cuboctahedron along one of its four "equilateral" hexagons.
%C Also as a(n)=(1/6)*(10*n^3-6*n^2+10*n), n>0: structured pentagonal anti-prism numbers (Cf. A100185 = structured anti-prisms); and structured tetragonal anti-diamond numbers (vertex structure 7) (Cf. A000447 = alternate vertex; A100188 = structured anti-diamonds). Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%D H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
%H T. D. Noe, <a href="/A096000/b096000.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = (1/2)*(Q(n) + 3n^2 + 3n + 1), where Q(n) are the cuboctahedral numbers, A005902.
%F G.f.: (1+6*x+3*x^2)/(1-x)^4. - _Paul Barry_, Oct 28 2006
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4), n>4. - _Wesley Ivan Hurt_, May 23 2015
%p A096000:=n->(n+1)*(5*n^2+7*n+3)/3; seq(A096000(n), n=0..50); # _Wesley Ivan Hurt_, Mar 11 2014
%t Table[(n + 1)(5n^2 + 7n + 3)/3, {n, 0, 50}] (* _Wesley Ivan Hurt_, Mar 11 2014 *)
%t CoefficientList[Series[(1 + 6 x + 3 x^2)/(1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 23 2015 *)
%o (PARI) a(n) = (1/3)*(n+1)*(5*n^2+7*n+3) \\ _Michel Marcus_, Jul 11 2013
%o (Magma) [(n+1)*(5*n^2+7*n+3)/3 : n in [0..50]]; // _Wesley Ivan Hurt_, May 23 2015
%Y Cf. A005902.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, in memory of Harold Scott MacDonald Coxeter [Feb 09 1907 - Mar 31 2003], May 08 2004
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