cp's OEIS Frontend

A005919 Number of points on surface of tricapped prism: a(n) = 7*n^2 + 2 for n > 0, a(0)=1.

%I A005919 M4607 #35 Nov 29 2024 18:02:58
%S A005919 1,9,30,65,114,177,254,345,450,569,702,849,1010,1185,1374,1577,1794,
%T A005919 2025,2270,2529,2802,3089,3390,3705,4034,4377,4734,5105,5490,5889,
%U A005919 6302,6729,7170,7625,8094,8577,9074,9585,10110,10649,11202,11769,12350,12945,13554
%N A005919 Number of points on surface of tricapped prism: a(n) = 7*n^2 + 2 for n > 0, a(0)=1.
%D A005919 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005919 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005919 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005919 B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
%H A005919 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A005919 From _Elmo R. Oliveira_, Nov 29 2024: (Start)
%F A005919 G.f.: (1 + x)*(1 + 5*x + x^2)/(1-x)^3.
%F A005919 E.g.f.: exp(x)*(7*x^2 + 7*x + 2) - 1.
%F A005919 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)
%p A005919 A005919:=-(z+1)*(z**2+5*z+1)/(z-1)**3; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A005919 Join[{1},7*Range[50]^2+2] (* or *) CoefficientList[Series[(-x^3-6x^2-6x-1)/(x-1)^3,{x,0,50}],x] (* _Harvey P. Dale_, Jan 13 2013 *)
%Y A005919 Cf. A206399.
%K A005919 nonn,easy
%O A005919 0,2
%A A005919 _N. J. A. Sloane_
%E A005919 More terms from _Erich Friedman_, Aug 08 2005