A146298 Difference between the cubes and 2*tetrahedral numbers; A000578(n) - 2*A000292(n).

0, -1, 0, 7, 24, 55, 104, 175, 272, 399, 560, 759, 1000, 1287, 1624, 2015, 2464, 2975, 3552, 4199, 4920, 5719, 6600, 7567, 8624, 9775, 11024, 12375, 13832, 15399, 17080, 18879, 20800, 22847, 25024, 27335, 29784, 32375, 35112, 37999, 41040, 44239, 47600, 51127

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = (n-2)*n*(2*n+1)/3.

G.f.: x*(-1+4*x+x^2)/(1-x)^4. - R. J. Mathar, Oct 31 2008

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012

E.g.f.: x*(-3 + 3*x + 2*x^2)*exp(x)/3. - Elmo R. Oliveira, Aug 22 2025

From Amiram Eldar, Aug 31 2025: (Start)

Sum_{n>=3} 1/a(n) = 12*log(2)/5 - 143/100.

Sum_{n>=3} (-1)^(n+1)/a(n) = 283/100 - 3*Pi/5 - 6*log(2)/5. (End)

Example

a(4) = (4 - 2)*4*(2*4 + 1)/3 = 2*4*3 = 24.

Mathematica

Table[(n-2)*n*(2*n+1)/3, {n, 0, 30}]
CoefficientList[Series[x*(-1+4*x+x^2)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 30 2012 *)

Prog

(SageMath) taylor( mul( x*(x^2+4*x-1)/(x-1)^4 for i in range(1, 2)), x, 0, 30) # Zerinvary Lajos, Jun 03 2009
(Magma) A000578:=func<i | i^3>; A000292:=func<i | i*(i+1)*(i+2)/6>; [A000578(n)-2*A000292(n): n in [0..30]]; // Bruno Berselli, Apr 07 2011

Crossrefs

Cf. A000292, A000578.

Sequence in context: A196116 A168668 A159225 * A079671 A212511 A100454

Adjacent sequences: A146295 A146296 A146297 * A146299 A146300 A146301

Keyword

sign,easy

Author

Jon Perry, Oct 29 2008

Extensions

Edited by Bruno Berselli, Apr 07 2011

Status

approved