A005905 Number of points on surface of truncated tetrahedron: a(n) = 14*n^2 + 2 for n > 0, a(0)=1.

1, 16, 58, 128, 226, 352, 506, 688, 898, 1136, 1402, 1696, 2018, 2368, 2746, 3152, 3586, 4048, 4538, 5056, 5602, 6176, 6778, 7408, 8066, 8752, 9466, 10208, 10978, 11776, 12602, 13456, 14338, 15248, 16186, 17152, 18146, 19168, 20218, 21296, 22402, 23536, 24698

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also sequence found by reading the segment (1,16) together with the line from 16, in the direction 16, 58, ... , in the square spiral whose vertices are the generalized enneagonal numbers A118277. - Omar E. Pol, Nov 05 2012

References

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

Links

• H. S. M. Coxeter, Polyhedral Numbers, in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
• 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. A118277, A206399.

Programs

Maple

A005905:=-(z+1)*(z**2+12*z+1)/(z-1)**3; # [Simon Plouffe in his 1992 dissertation.]

Mathematica

a[0] = 1; a[n_] := 14 n^2 + 2; Table[a[n], {n, 0, 50}] (* Wesley Ivan Hurt, Mar 04 2014 *)

PARI

a(n) = if (n==0, 1, 14*n^2+2); \\ Michel Marcus, Mar 04 2014

Formula

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

G.f.: (1 + x)*(1 + 12*x + x^2)/(1-x)^3.

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

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

Extensions

More terms from Michel Marcus, Mar 04 2014