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A005906
Truncated tetrahedral numbers: a(n) = (1/6)*(n+1)*(23*n^2 + 19*n + 6).
(Formerly M5002)
3
1, 16, 68, 180, 375, 676, 1106, 1688, 2445, 3400, 4576, 5996, 7683, 9660, 11950, 14576, 17561, 20928, 24700, 28900, 33551, 38676, 44298, 50440, 57125, 64376, 72216, 80668, 89755, 99500, 109926, 121056, 132913, 145520, 158900, 173076
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
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.
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
A005906:=n->(1/6)*(n+1)*(23*n^2+19*n+6): seq(A005906(n), n=0..80); # Wesley Ivan Hurt, Nov 04 2017
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
Cf. A000292.
Sequence in context: A344600 A271913 A178574 * A247663 A235643 A297886
KEYWORD
nonn,easy
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