OFFSET
0,2
COMMENTS
a(n) is the number of 4-element subsets of {-n,...,0,...,n} having sum n. - Clark Kimberling, Apr 05 2012
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus (Springer imprint), New York: Springer-Verlag, 1996, ch. 2, pp. 46-47. (In the formula it should read Tet_{3*n-2} not Tet_{3*n-3}).
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
John Cerkan, 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.
Eric Weisstein's World of Mathematics, Truncated Tetrahedral Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = binomial(3*n, 3) - 4*binomial(n+1, 3) = n*(23*n^2 -27*n +10)/6.
a(n-1) = Tet(3*n-2) - 4*Tet(n-1) = (1/6)*n*(23*n^2 - 27*n + 10), n >= 1, with Tet(n) = A000292(n). See the Conway-Guy reference, with a corrected misprint. - Wolfdieter Lang, Jan 09 2017
From G. C. Greubel, Nov 04 2017: (Start)
G.f.: x*(1 + 12*x + 10*x^2)/(1 - x)^4.
E.g.f.: (x/6)*(6 + 42*x + 23*x^2)*exp(x). (End)
MAPLE
A005906:=(1+12*z+10*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(1/6) (n + 1) (23 n^2 + 19 n + 6), {n, 0, 35}] (* or *)
Table[Binomial[3 n, 3] - 4 Binomial[n + 1, 3], {n, 36}] (* Michael De Vlieger, Mar 10 2016 *)
PROG
(PARI) a(n)=(n+1)*(23*n^2+19*n+6)/6 \\ Charles R Greathouse IV, Feb 22 2017
(Magma) [n*(23*n^2 -27*n +10)/6: n in [0..50]]; // G. C. Greubel, Nov 04 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 20 1999
Corrected by T. D. Noe, Nov 07 2006
STATUS
approved