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

1, 9, 30, 65, 114, 177, 254, 345, 450, 569, 702, 849, 1010, 1185, 1374, 1577, 1794, 2025, 2270, 2529, 2802, 3089, 3390, 3705, 4034, 4377, 4734, 5105, 5490, 5889, 6302, 6729, 7170, 7625, 8094, 8577, 9074, 9585, 10110, 10649, 11202, 11769, 12350, 12945, 13554

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A206399.

Programs

Maple

A005919:=-(z+1)*(z**2+5*z+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)

Formula

From Elmo R. Oliveira, Nov 29 2024: (Start)

G.f.: (1 + x)*(1 + 5*x + x^2)/(1-x)^3.

E.g.f.: exp(x)*(7*x^2 + 7*x + 2) - 1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)

Extensions

More terms from Erich Friedman, Aug 08 2005