A005912 Truncated cube numbers.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Programs

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

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 *)

PARI

a(n)=(3*n+1)^3-8*(n)*(n+1)*(n+2)/6 \\ Charles R Greathouse IV, Feb 10 2017

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

Extensions

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