A005906 Truncated tetrahedral numbers: a(n) = (1/6)*(n+1)*(23*n^2 + 19*n + 6). (Formerly M5002)

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

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
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

Adjacent sequences: A005903 A005904 A005905 * A005907 A005908 A005909

Keyword

nonn,easy

Author

N. J. A. Sloane

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