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A096000 Cupolar numbers: a(n) = (n+1)*(5*n^2 + 7*n + 3)/3.

%I A096000 #38 Aug 14 2025 01:08:33
%S A096000 1,10,37,92,185,326,525,792,1137,1570,2101,2740,3497,4382,5405,6576,
%T A096000 7905,9402,11077,12940,15001,17270,19757,22472,25425,28626,32085,
%U A096000 35812,39817,44110,48701,53600,58817,64362,70245,76476,83065,90022,97357,105080,113201,121730
%N A096000 Cupolar numbers: a(n) = (n+1)*(5*n^2 + 7*n + 3)/3.
%C A096000 Number of equal balls that will fill a triangular cupola, formed by splitting a cuboctahedron along one of its four "equilateral" hexagons.
%C A096000 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 A096000 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 A096000 T. D. Noe, <a href="/A096000/b096000.txt">Table of n, a(n) for n = 0..1000</a>
%H A096000 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 A096000 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A096000 a(n) = (1/2)*(Q(n) + 3*n^2 + 3*n + 1), where Q(n) are the cuboctahedral numbers, A005902.
%F A096000 G.f.: (1+6*x+3*x^2)/(1-x)^4. - _Paul Barry_, Oct 28 2006
%F A096000 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - _Wesley Ivan Hurt_, May 23 2015
%F A096000 E.g.f.: exp(x)*(3 + 27*x + 27*x^2 + 5*x^3)/3. - _Elmo R. Oliveira_, Aug 11 2025
%p A096000 A096000:=n->(n+1)*(5*n^2+7*n+3)/3; seq(A096000(n), n=0..50); # _Wesley Ivan Hurt_, Mar 11 2014
%t A096000 Table[(n + 1)(5n^2 + 7n + 3)/3, {n, 0, 50}] (* _Wesley Ivan Hurt_, Mar 11 2014 *)
%t A096000 CoefficientList[Series[(1 + 6 x + 3 x^2)/(1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 23 2015 *)
%o A096000 (PARI) a(n) = (1/3)*(n+1)*(5*n^2+7*n+3) \\ _Michel Marcus_, Jul 11 2013
%o A096000 (Magma) [(n+1)*(5*n^2+7*n+3)/3 : n in [0..50]]; // _Wesley Ivan Hurt_, May 23 2015
%Y A096000 Cf. A000447, A005902, A100145, A100185, A100188.
%K A096000 nonn,easy
%O A096000 0,2
%A A096000 _N. J. A. Sloane_, in memory of Harold Scott MacDonald Coxeter [Feb 09 1907 - Mar 31 2003], May 08 2004