A005912 Truncated cube numbers. (Formerly M5312)

1, 56, 311, 920, 2037, 3816, 6411, 9976, 14665, 20632, 28031, 37016, 47741, 60360, 75027, 91896, 111121, 132856, 157255, 184472, 214661, 247976, 284571, 324600, 368217, 415576, 466831, 522136, 581645, 645512, 713891, 786936, 864801, 947640, 1035607, 1128856

Offset

0,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, 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.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Formula

a(n) = (3*n+1)^3 - 8*(n)*(n+1)*(n+2)/6 = (77/3)*n^3 + 23*n^2 + (19/3)*n + 1.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=56, a(2)=311, a(3)=920. - Harvey P. Dale, Aug 14 2011

Maple

A005912:=(1+52*z+93*z**2+8*z**3)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[(3n+1)^3-8(n)(n+1)(n+2)/6, {n, 0, 30}] (* or *) LinearRecurrence[ {4, -6, 4, -1}, {1, 56, 311, 920}, 30] (* Harvey P. Dale, Aug 14 2011 *)

Prog

(Haskell)
a005912 n = (n * (n * (77 * n + 69) + 19) + 3) `div` 3 :: Integer
-- Reinhard Zumkeller, Aug 09 2014
(Magma) [(3*n+1)^3-8*(n)*(n+1)*(n+2)/6: n in [0..40]] // Vincenzo Librandi, Aug 09 2014
(PARI) a(n)=(3*n+1)^3-8*(n)*(n+1)*(n+2)/6 \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Sequence in context: A110554 A253922 A220015 * A264581 A220202 A104677

Adjacent sequences: A005909 A005910 A005911 * A005913 A005914 A005915

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

Status

approved