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%I #27 Sep 08 2022 08:45:38
%S 0,-1,0,7,24,55,104,175,272,399,560,759,1000,1287,1624,2015,2464,2975,
%T 3552,4199,4920,5719,6600,7567,8624,9775,11024,12375,13832,15399,17080
%N Difference between the cubes and 2*tetrahedral numbers; A000578(n) - 2*A000292(n).
%C Can be visualized as layering a cube up from a corner. Eventually the series of triangular numbers is truncated. So 7 = 10-3 (the corners are removed), 24 = 15+15-3-3 and 55 = 21+28+21-3-9-3.
%H Vincenzo Librandi, <a href="/A146298/b146298.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) = (n-2)*n*(2n+1)/3.
%F G.f.: x*(-1+4*x+x^2)/(1-x)^4. - _R. J. Mathar_, Oct 31 2008
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 30 2012
%e a(4) = (4 - 2)*4*(2*4 + 1)/3 = 2*4*3 = 24.
%t Table[(n-2)*n*(2*n+1)/3,{n,0,30}]
%t CoefficientList[Series[x*(-1+4*x+x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 30 2012 *)
%o (Sage) 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
%o (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
%Y Cf. A000292, A000578.
%K sign,easy
%O 0,4
%A _Jon Perry_, Oct 29 2008
%E Edited by _Bruno Berselli_, Apr 07 2011
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