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Exponential self-convolution of Pell numbers.
3

%I #21 Sep 08 2022 08:44:35

%S 0,0,2,12,64,320,1568,7616,36864,178176,860672,4156416,20070400,

%T 96911360,467935232,2259402752,10909384704,52675215360,254338531328,

%U 1228055248896,5929575645184,28630524624896

%N Exponential self-convolution of Pell numbers.

%H Vincenzo Librandi, <a href="/A006646/b006646.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4,-8).

%F a(n) = ((2+sqrt(8))^n+(2-sqrt(8))^n-2^(n+1))/8; E.g.f.: exp(2x)(sinh(sqrt(2)*x))^2/2 = (exp(x)*sinh(sqrt(2)*x)/sqrt(2))^2. - _Paul Barry_, May 16 2003

%F G.f.: 2*x^2 / ( (2*x-1)*(4*x^2+4*x-1) ). - _R. J. Mathar_, Nov 24 2012

%F a(n) = 2^(n-3)*(A002203(n) - 2). - _Vladimir Reshetnikov_, Oct 07 2016

%t Table[2^(n-3)*(LucasL[n, 2] - 2), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 07 2016 *)

%o (Magma) [Floor(((2+Sqrt(8))^n+(2-Sqrt(8))^n-2^(n+1))/8): n in [0..30] ]; // _Vincenzo Librandi_, Aug 20 2011

%Y Cf. A000129, A002203, A006668.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_