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%I #25 Sep 08 2022 08:45:56
%S 1,0,64,665,3431,12194,34462,83187,178957,352484,647516,1124045,
%T 1861939,2964870,4564666,6825959,9951257,14186312,19825912,27219969,
%U 36780031,48986090,64393814,83642075,107460901
%N a(n) = n^6 - a(n-1), a(0)=1.
%C a(n)+a(n-1) is a perfect 6th power, hence a perfect square and a perfect cube.
%H Vincenzo Librandi, <a href="/A187620/b187620.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1).
%F a(n) = (-1)^n + n*(3-5*n^2+3*n^4+n^5)/2. - _R. J. Mathar_, Mar 15 2011
%F a(n) = (-1)^n+A152725(n). - _R. J. Mathar_, Mar 15 2011
%F G.f. ( -1-78*x^2-267*x^3-337*x^4-36*x^5-8*x^6+x^7+6*x ) / ( (1+x)*(x-1)^7 ). - _R. J. Mathar_, Mar 15 2011
%t CoefficientList[Series[(- 1 - 78 x^2 - 267 x^3 - 337 x^4 - 36 x^5 - 8 x^6 + x^7 + 6 x)/((1 + x) (x - 1)^7), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 04 2013 *)
%t LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{1,0,64,665,3431,12194,34462,83187},30] (* _Harvey P. Dale_, Apr 30 2020 *)
%o (Magma) [(-1)^n + n*(3-5*n^2+3*n^4+n^5)/2: n in [0..30]]; // _Vincenzo Librandi_, Oct 04 2013
%Y Cf. A152725.
%K nonn,easy
%O 0,3
%A _Vincenzo Librandi_, Mar 12 2011
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