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%I M5002 #71 Sep 08 2022 08:44:34
%S 1,16,68,180,375,676,1106,1688,2445,3400,4576,5996,7683,9660,11950,
%T 14576,17561,20928,24700,28900,33551,38676,44298,50440,57125,64376,
%U 72216,80668,89755,99500,109926,121056,132913,145520,158900,173076
%N Truncated tetrahedral numbers: a(n) = (1/6)*(n+1)*(23*n^2 + 19*n + 6).
%C a(n) is the number of 4-element subsets of {-n,...,0,...,n} having sum n. - _Clark Kimberling_, Apr 05 2012
%D 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}).
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H John Cerkan, <a href="/A005906/b005906.txt">Table of n, a(n) for n = 0..10000</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TruncatedTetrahedralNumber.html">Truncated Tetrahedral Number.</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = binomial(3*n, 3) - 4*binomial(n+1, 3) = n*(23*n^2 -27*n +10)/6.
%F 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
%F From _G. C. Greubel_, Nov 04 2017: (Start)
%F G.f.: x*(1 + 12*x + 10*x^2)/(1 - x)^4.
%F E.g.f.: (x/6)*(6 + 42*x + 23*x^2)*exp(x). (End)
%p A005906:=(1+12*z+10*z**2)/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p A005906:=n->(1/6)*(n+1)*(23*n^2+19*n+6): seq(A005906(n), n=0..80); # _Wesley Ivan Hurt_, Nov 04 2017
%t Table[(1/6) (n + 1) (23 n^2 + 19 n + 6), {n, 0, 35}] (* or *)
%t Table[Binomial[3 n, 3] - 4 Binomial[n + 1, 3], {n, 36}] (* _Michael De Vlieger_, Mar 10 2016 *)
%o (PARI) a(n)=(n+1)*(23*n^2+19*n+6)/6 \\ _Charles R Greathouse IV_, Feb 22 2017
%o (Magma) [n*(23*n^2 -27*n +10)/6: n in [0..50]]; // _G. C. Greubel_, Nov 04 2017
%Y Cf. A000292.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 20 1999
%E Corrected by _T. D. Noe_, Nov 07 2006
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