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Exponential self-convolution of Jacobsthal numbers (divided by 2).
5

%I #33 May 08 2024 05:40:34

%S 0,0,1,3,15,55,231,903,3655,14535,58311,232903,932295,3727815,

%T 14913991,59650503,238612935,954429895,3817763271,15270965703,

%U 61084037575,244335800775,977343902151,3909374210503,15637499638215,62549992960455

%N Exponential self-convolution of Jacobsthal numbers (divided by 2).

%H Vincenzo Librandi, <a href="/A084152/b084152.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).

%F a(n) = (4^n - 2 + (-2)^n)/18.

%F G.f.: x^2/((1-x)*(1+2*x)*(1-4*x)).

%F a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3).

%F E.g.f.: (exp(2*x) - exp(-x))^2/18 = (exp(4*x) - 2*exp(x) + exp(-x))/18.

%F Binomial transform of 0, 0, 1, 0, 9, 0, 81, ... .

%F a(n) = A001045(n)*A078008(n)/2.

%F a(n) = floor(2^n/3)ceiling(2^n/3)/2. - _Paul Barry_, Apr 28 2004

%t Join[{a=0,b=0},Table[c=2*b+8*a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011*)

%t LinearRecurrence[{3,6,-8},{0,0,1},30] (* _Harvey P. Dale_, Nov 11 2011 *)

%o (Magma) [(4^n-2+(-2)^n)/18: n in [0..35]]; // _Vincenzo Librandi_, Jul 05 2011

%o (SageMath) [(4^n-2+(-2)^n)/18 for n in range(41)] # _G. C. Greubel_, Oct 11 2022

%Y Cf. A001045, A078008, A084153.

%Y Except for initial terms, same as A015249 and A084175.

%K easy,nonn

%O 0,4

%A _Paul Barry_, May 16 2003