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

%I A060999 #36 Jun 10 2025 08:42:06
%S A060999 0,1,3,7,14,24,38,57,81,111,148,192,244,305,375,455,546,648,762,889,
%T A060999 1029,1183,1352,1536,1736,1953,2187,2439,2710,3000,3310,3641,3993,
%U A060999 4367,4764,5184,5628,6097,6591,7111,7658,8232,8834,9465,10125,10815,11536
%N A060999 Nearest integer to (n+1)^3/9.
%C A060999 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
%H A060999 Harry J. Smith, <a href="/A060999/b060999.txt">Table of n, a(n) for n=0..1000</a>
%H A060999 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A060999 G.f.: x*(1+x^2)/((1-x)^3*(1-x^3)).
%F A060999 G.f.: x * (1 - x^4) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)).
%F A060999 G.f.: ( (1 + 4*x + x^2) / (1 - x)^4 - 1 / (1 + x + x^2) ) / 9.
%F A060999 From _Michael Somos_, Aug 12 2009: (Start)
%F A060999 Euler transform of length 4 sequence [ 3, 1, 1, -1].
%F A060999 a(-2-n) = -a(n). (End)
%F A060999 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
%F A060999 From _Hoang Xuan Thanh_, Jun 03 2025: (Start)
%F A060999 a(n) = floor(((n+1)^3+1)/9).
%F A060999 For n>0: a(n) = A266769(2n-2). (End)
%e A060999 x + 3*x^2 + 7*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 57*x^7 + 81*x^8 + ...
%t A060999 Table[Floor[(n+1)^3/9+1/2],{n,0,50}] (* _Harvey P. Dale_, Jan 20 2013 *) (* or *)
%t A060999 LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 1, 3, 7, 14, 24}, 47] (* _Georg Fischer_, Oct 13 2020 *)
%o A060999 (PARI) a(n) = { round((n + 1)^3/9) } \\ _Harry J. Smith_, Jul 16 2009
%o A060999 (PARI) {a(n) = n++; (n^3 - kronecker(-3, n)) / 9} /* _Michael Somos_, Aug 12 2009 */
%Y A060999 Cf. A266769, A008763.
%K A060999 nonn,easy
%O A060999 0,3
%A A060999 _N. J. A. Sloane_, May 14 2001