A060999 Nearest integer to (n+1)^3/9.

0, 1, 3, 7, 14, 24, 38, 57, 81, 111, 148, 192, 244, 305, 375, 455, 546, 648, 762, 889, 1029, 1183, 1352, 1536, 1736, 1953, 2187, 2439, 2710, 3000, 3310, 3641, 3993, 4367, 4764, 5184, 5628, 6097, 6591, 7111, 7658, 8232, 8834, 9465, 10125, 10815, 11536

Offset: 0

Author

N. J. A. Sloane, May 14 2001

Keywords

• nonn
• easy

Comments

a(n) is also the number of ways to award 4n+5 bonuses to 4 teams: first, second, third and fourth satisfying 1st > 2nd > 3rd > 4th and 1st + 4th < 2nd + 3rd. - Hoang Xuan Thanh, Jun 03 2025

Examples

			x + 3*x^2 + 7*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 57*x^7 + 81*x^8 + ...
		

Links

• Harry J. Smith, Table of n, a(n) for n=0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A266769, A008763.

Programs

Mathematica

Table[Floor[(n+1)^3/9+1/2],{n,0,50}] (* Harvey P. Dale, Jan 20 2013 *) (* or *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 1, 3, 7, 14, 24}, 47] (* Georg Fischer, Oct 13 2020 *)

PARI

a(n) = { round((n + 1)^3/9) } \\ Harry J. Smith, Jul 16 2009

PARI

{a(n) = n++; (n^3 - kronecker(-3, n)) / 9} /* Michael Somos, Aug 12 2009 */

Formula

G.f.: x*(1+x^2)/((1-x)^3*(1-x^3)).

G.f.: x * (1 - x^4) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)).

G.f.: ( (1 + 4*x + x^2) / (1 - x)^4 - 1 / (1 + x + x^2) ) / 9.

From Michael Somos, Aug 12 2009: (Start)

Euler transform of length 4 sequence [ 3, 1, 1, -1].

a(-2-n) = -a(n). (End)

E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(1 + 7*x + 6*x^2 + x^3) - 3*cos(sqrt(3)*x/2)+ sqrt(3)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Sep 24 2024

From Hoang Xuan Thanh, Jun 03 2025: (Start)

a(n) = floor(((n+1)^3+1)/9).

For n>0: a(n) = A266769(2n-2). (End)