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A084137 Binomial transform of A084136.

%I A084137 #15 Oct 14 2022 08:55:36
%S A084137 1,2,8,32,144,672,3200,15360,73984,356864,1722368,8314880,40144896,
%T A084137 193830912,935886848,4518838272,21818834944,105350561792,508677324800,
%U A084137 2456111022080,11859152338944,57261051346944,276480810549248
%N A084137 Binomial transform of A084136.
%C A084137 Exponential self-convolution of companion Pell numbers (A002203), divided by 4. - _Vladimir Reshetnikov_, Oct 07 2016
%H A084137 Michael De Vlieger, <a href="/A084137/b084137.txt">Table of n, a(n) for n = 0..1463</a>
%H A084137 Sergio Falcon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.96-106">Half self-convolution of the k-Fibonacci sequence</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
%H A084137 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4,-8).
%F A084137 G.f.: (1-4*x)/((1-2*x)*(1-4*x-4*x^2)).
%F A084137 E.g.f.: exp(2*x)*cosh(sqrt(2)*x)^2 = (exp(x)*cosh(sqrt(2)*x))^2.
%F A084137 a(n) = ((2+sqrt(8))^n + (2-sqrt(8))^n + 2^(n+1))/4.
%F A084137 a(n) = (A084128(n) + 2^n)/2.
%F A084137 a(n) = 2^(n-2)*(2 + A002203(n)). - _Vladimir Reshetnikov_, Oct 07 2016
%F A084137 a(n) = 6*a(n-1) - 4*a(n-2) - 8*a(n-3). - _G. C. Greubel_, Oct 13 2022
%t A084137 Table[2^(n-2)*(2+LucasL[n,2]), {n,0,20}] (* _Vladimir Reshetnikov_, Oct 07 2016 *)
%o A084137 (PARI) a(n)=if(n<0,0,polsym(4+4*x-x^2,n)[n+1]/4+2^(n-1))
%o A084137 (Magma)
%o A084137 A002203:= func< n | Round((1+Sqrt(2))^n + (1-Sqrt(2))^n) >;
%o A084137 [2^(n-2)*(2+A002203(n)): n in [0..40]]; // _G. C. Greubel_, Oct 13 2022
%o A084137 (SageMath) [2^(n-2)*(2+lucas_number2(n, 2, -1)) for n in range(41)] # _G. C. Greubel_, Oct 13 2022
%Y A084137 Cf. A002203, A084128, A084136.
%K A084137 easy,nonn
%O A084137 0,2
%A A084137 _Paul Barry_, May 16 2003