OFFSET
0,2
COMMENTS
Equivalently, the number of points of the surface of an icosahedron of equivalent side length, constructed by centered 20 triangular numbers. - Benjamin W P Cornish, Nov 12 2025
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 137-138.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
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. 24 (1985), 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (1+x)*(1+28*x+x^2)/(1-x)^3. - Simon Plouffe (see MAPLE line)
Sum_{n>=0} 1/a(n) = 3/4 + Pi*sqrt(15)*coth(Pi/sqrt 15)/60 = 1.052567... - R. J. Mathar, Apr 27 2024
From Elmo R. Oliveira, Sep 07 2025: (Start)
E.g.f.: 2*exp(x)*(15*x^2 + 15*x + 1) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
MAPLE
A005903:=-(z+1)*(z**2+28*z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation.
MATHEMATICA
Join[{1}, 30 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 32, 122, 272}, 50] (* Harvey P. Dale, Jan 23 2026 *)
PROG
(PARI) a(n) = if (n==0, 1, 30*n^2+2); \\ Michel Marcus, Mar 04 2014
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved