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A187272 a(n) = (n/4)*2^(n/2)*((1+sqrt(2))^2 + (-1)^n*(1-sqrt(2))^2).

%I A187272 #51 Feb 18 2024 12:05:17
%S A187272 0,2,6,12,24,40,72,112,192,288,480,704,1152,1664,2688,3840,6144,8704,
%T A187272 13824,19456,30720,43008,67584,94208,147456,204800,319488,442368,
%U A187272 688128,950272,1474560,2031616,3145728,4325376,6684672,9175040,14155776,19398656,29884416,40894464,62914560,85983232
%N A187272 a(n) = (n/4)*2^(n/2)*((1+sqrt(2))^2 + (-1)^n*(1-sqrt(2))^2).
%H A187272 Vincenzo Librandi, <a href="/A187272/b187272.txt">Table of n, a(n) for n = 0..1000</a>
%H A187272 R. Kemp, <a href="http://dx.doi.org/10.1016/0012-365X(82)90123-6">On the number of words in the language {w in Sigma* | w = w^R }^2</a>, Discrete Math., 40 (1982), 225-234. See Lemma 6 (p. 228).
%H A187272 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-4).
%F A187272 From _Bruno Berselli_, Mar 22 2011: (Start)
%F A187272 G.f.: 2*x*(1+x)*(1+2*x)/(1-2*x^2)^2.
%F A187272 a(n)/a(n-2) = 2*n/(n-2). (End)
%F A187272 a(2*n) = 3*n*2^n, a(2*n+1) = (2*n+1)*2^(n+1). - _Andrew Howroyd_, Mar 28 2016
%p A187272 R:=(b,n)-> expand(simplify( (n/4)*b^(n/2)*((1+sqrt(b))^2+(-1)^n*(1-sqrt(b))^2) ));
%p A187272 [seq(R(2,n),n=0..100)];
%t A187272 CoefficientList[Series[2 x (1 + x) (1 + 2 x) / (1 - 2 x^2)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)
%o A187272 (PARI) x='x+O('x^30); concat([0], Vec(2*x*(1+x)*(1+2*x)/(1-2*x^2)^2)) \\ _G. C. Greubel_, Nov 28 2017
%o A187272 (Magma) [Round((n/4)*2^(n/2)*((1+Sqrt(2))^2 + (-1)^n*(1-Sqrt(2))^2)): n in [0..30]]; // _G. C. Greubel_, Nov 28 2017
%o A187272 (Python)
%o A187272 def A187272(n): return (n<<(n+1>>1) if n&1 else 3*n<<(n-2>>1)) if n else 0 # _Chai Wah Wu_, Feb 18 2024
%Y A187272 Cf. A187273, A187274, A187275.
%Y A187272 Cf. A007055, A007056, A007057, A007058.
%K A187272 nonn,easy
%O A187272 0,2
%A A187272 _N. J. A. Sloane_, Mar 07 2011