A005919 Number of points on surface of tricapped prism: a(n) = 7*n^2 + 2 for n > 0, a(0)=1. (Formerly M4607)

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

0,2

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 416.

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

Links

Table of n, a(n) for n=0..44.

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).

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)

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 *)

Crossrefs

Cf. A206399.

Sequence in context: A195319 A073399 A225275 * A084370 A000439 A002414

Adjacent sequences: A005916 A005917 A005918 * A005920 A005921 A005922

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Erich Friedman, Aug 08 2005

Status

approved