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A084152 Exponential self-convolution of Jacobsthal numbers (divided by 2).

%I A084152 #37 Apr 25 2025 09:47:53
%S A084152 0,0,1,3,15,55,231,903,3655,14535,58311,232903,932295,3727815,
%T A084152 14913991,59650503,238612935,954429895,3817763271,15270965703,
%U A084152 61084037575,244335800775,977343902151,3909374210503,15637499638215,62549992960455
%N A084152 Exponential self-convolution of Jacobsthal numbers (divided by 2).
%H A084152 Vincenzo Librandi, <a href="/A084152/b084152.txt">Table of n, a(n) for n = 0..500</a>
%H A084152 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).
%F A084152 a(n) = (4^n - 2 + (-2)^n)/18.
%F A084152 G.f.: x^2/((1-x)*(1+2*x)*(1-4*x)).
%F A084152 a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3).
%F A084152 E.g.f.: (exp(2*x) - exp(-x))^2/18 = (exp(4*x) - 2*exp(x) + exp(-x))/18.
%F A084152 Binomial transform of 0, 0, 1, 0, 9, 0, 81, ... .
%F A084152 a(n) = A001045(n)*A078008(n)/2.
%F A084152 a(n) = floor(2^n/3)ceiling(2^n/3)/2. - _Paul Barry_, Apr 28 2004
%t A084152 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 A084152 LinearRecurrence[{3,6,-8},{0,0,1},30] (* _Harvey P. Dale_, Nov 11 2011 *)
%o A084152 (Magma) [(4^n-2+(-2)^n)/18: n in [0..35]]; // _Vincenzo Librandi_, Jul 05 2011
%o A084152 (SageMath) [(4^n-2+(-2)^n)/18 for n in range(41)] # _G. C. Greubel_, Oct 11 2022
%o A084152 (Python)
%o A084152 def A084152(n): return ((m:=1<<n)|1)//3*((m>>1|1)//3) # _Chai Wah Wu_, Apr 25 2025
%Y A084152 Cf. A001045, A078008, A084153.
%Y A084152 Except for initial terms, same as A015249 and A084175.
%K A084152 easy,nonn
%O A084152 0,4
%A A084152 _Paul Barry_, May 16 2003