|
%I #17 Oct 24 2022 11:12:06
%S 0,2,6,12,22,34,48,66,86,108,134,162,192,226,262,300,342,386,432,482,
%T 534,588,646,706,768,834,902,972,1046,1122,1200,1282,1366,1452,1542,
%U 1634,1728,1826,1926,2028,2134,2242,2352,2466,2582,2700,2822,2946,3072
%N The 3rd Witt transform of A040000.
%C The 2nd Witt transform of A040000 is represented by A042964.
%H Vincenzo Librandi, <a href="/A147623/b147623.txt">Table of n, a(n) for n = 0..1000</a>
%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F G.f.: 2*x*(1+x)*(1+x^2)/((1-x)^3*(1+x+x^2)).
%F a(n) = 2*A071619(n).
%F From _G. C. Greubel_, Oct 24 2022: (Start)
%F a(n) = 4*(2 - 2*n + n^2) - a(n-1) - a(n-2).
%F a(n) = 2*(2*(1 + 3*n^2) - (2*A049347(n) + A049347(n-1)))/9. (End)
%t CoefficientList[Series[2x(1+x)(1 +x^2)/((1-x)^3 (1+x+x^2)), {x,0,40}], x] (* _Vincenzo Librandi_, Dec 14 2012 *)
%t LinearRecurrence[{2,-1,1,-2,1},{0,2,6,12,22},50] (* _Harvey P. Dale_, Jul 04 2021 *)
%o (Magma) [n le 2 select 1+(-1)^n else 4*(1+(n-2)^2) - Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Oct 24 2022
%o (SageMath) [2*(2*(1+3*n^2) -(2*chebyshev_U(n, -1/2) +chebyshev_U(n-1, -1/2)))/9 for n in range(41)] # _G. C. Greubel_, Oct 24 2022
%Y Cf. A040000, A042964, A049347, A071619.
%K nonn,easy
%O 0,2
%A _R. J. Mathar_, Nov 08 2008
|
