OFFSET
0,2
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 117.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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 (4,-6,4,-1).
FORMULA
a(n) = (3*n+1)^3 - 8*n*(n+1)*(n+2)/6 = (77/3)*n^3 + 23*n^2 + (19/3)*n + 1.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=56, a(2)=311, a(3)=920. - Harvey P. Dale, Aug 14 2011
E.g.f.: exp(x)*(3 + 165*x + 300*x^2 + 77*x^3)/3. - Elmo R. Oliveira, Sep 07 2025
MAPLE
A005912:=(1+52*z+93*z**2+8*z**3)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(3n+1)^3-8n(n+1)(n+2)/6, {n, 0, 30}] (* Harvey P. Dale, Aug 14 2011 *)
(* Alternative: *)
LinearRecurrence[ {4, -6, 4, -1}, {1, 56, 311, 920}, 30] (* Harvey P. Dale, Aug 14 2011 *)
PROG
(Haskell)
a005912 n = (n * (n * (77 * n + 69) + 19) + 3) `div` 3 :: Integer
-- Reinhard Zumkeller, Aug 09 2014
(Magma) [(3*n+1)^3-8*n*(n+1)*(n+2)/6: n in [0..40]]; // Vincenzo Librandi, Aug 09 2014
(PARI) a(n)=(3*n+1)^3-8*n*(n+1)*(n+2)/6 \\ Charles R Greathouse IV, Feb 10 2017
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999
STATUS
approved