A005906 Truncated tetrahedral numbers: a(n) = (1/6)*(n+1)*(23*n^2 + 19*n + 6).
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
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).
Crossrefs
Cf. A000292.
Programs
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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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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
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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 *) LinearRecurrence[{4,-6,4,-1},{1,16,68,180},40] (* Harvey P. Dale, Dec 31 2024 *) -
PARI
a(n)=(n+1)*(23*n^2+19*n+6)/6 \\ Charles R Greathouse IV, Feb 22 2017
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)
Extensions
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 20 1999
Corrected by T. D. Noe, Nov 07 2006
Comments