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%I #50 Sep 08 2022 08:45:38
%S 0,4,17,45,94,170,279,427,620,864,1165,1529,1962,2470,3059,3735,4504,
%T 5372,6345,7429,8630,9954,11407,12995,14724,16600,18629,20817,23170,
%U 25694,28395,31279,34352,37620,41089,44765,48654,52762,57095,61659
%N The arithmetic mean of the n-th and (n+1)-st cubes, rounded down.
%C The terms of this sequence relate to intervals between cubes in the same fashion as terms of A002378 are related to intervals between squares.
%H Vincenzo Librandi, <a href="/A147656/b147656.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = floor((A000578(n) + A000578(n+1))/2).
%F From _R. J. Mathar_, Nov 11 2008: (Start)
%F a(n) = A000578(n) + A045943(n) = n*(2n^2+3n+3)/2.
%F G.f.: x*(4+x+x^2)/(1-x)^4. (End)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, May 06 2012
%F a(n) = A027480(n) + A006003(n). - _Bruce J. Nicholson_, Jun 03 2018
%F From _A.H.M. Smeets_, Sep 10 2018: (Start)
%F a(n) = Sum_{k=0..n-1} (n+1)^2-k for n >= 0 with empty domain of summation for n = 0.
%F a(n) = n*(n+1)^2 - n*(n-1)/2 for n >= 0.
%F Lim_{n -> inf} a(n-1)/n^3 = 1. (End)
%F E.g.f.: exp(x)*(8*x + 9*x^2 + 2*x^3)/2. - _Stefano Spezia_, Sep 12 2018
%F a(n) = A081435(n)-1. - _R. J. Mathar_, Sep 14 2018
%p seq(coeff(series(x*(x^2+x+4)/(1-x)^4,x,n+1), x, n), n = 0 .. 40); # _Muniru A Asiru_, Sep 11 2018
%t Table[(n^3+(n+1)^3-1)/2,{n,0,70}] (* _Vladimir Joseph Stephan Orlovsky_, May 04 2011 *)
%o (PARI) j=[];for (n=0,40,j=concat(j,n^3+floor(((n+1)^3 - n^3)/2)));j
%o (PARI) a(n) = n*(2*n^2+3*n+3)/2; \\ _Altug Alkan_, Sep 20 2018
%o (Magma) I:=[0, 4, 17, 45]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, May 06 2012
%Y Cf. A000578.
%Y Cf. A027480, A006003.
%K nonn,easy
%O 0,2
%A _Alexander R. Povolotsky_, Nov 09 2008
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