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A005906
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Truncated tetrahedral numbers: a(n) = (1/6)*(n+1)*(23*n^2 + 19*n + 6).
(Formerly M5002)
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3
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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
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of 4-element subsets of {-n,...,0,...,n} having sum n. - Clark Kimberling, Apr 05 2012
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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)
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(Magma) [n*(23*n^2 -27*n +10)/6: n in [0..50]]; // G. C. Greubel, Nov 04 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 20 1999
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STATUS
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approved
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