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A014335 Exponential convolution of Fibonacci numbers with themselves (divided by 2).

%I A014335 #49 Jan 06 2023 02:07:17
%S A014335 0,0,1,3,11,35,115,371,1203,3891,12595,40755,131891,426803,1381171,
%T A014335 4469555,14463795,46805811,151466803,490156851,1586180915,5132989235,
%U A014335 16610702131,53753361203,173949530931,562912506675,1821623137075,5894896300851,19076285150003
%N A014335 Exponential convolution of Fibonacci numbers with themselves (divided by 2).
%C A014335 It can be noticed that A014335/A011782 is an "autosequence", that is a sequence which is identical to its inverse binomial transform, except for alternating signs. - _Jean-François Alcover_, Jun 15 2016
%H A014335 Alois P. Heinz, <a href="/A014335/b014335.txt">Table of n, a(n) for n = 0..1000</a>
%H A014335 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-4).
%F A014335 a(n) = A014334(n)/2.
%F A014335 G.f.: x^2/((1-x)*(1-2*x-4*x^2)). - _Vladeta Jovovic_, Mar 05 2003
%F A014335 E.g.f.: exp(x)*(cosh(sqrt(5)*x)-1)/5. - _Vladeta Jovovic_, Sep 01 2004
%F A014335 From _Benoit Cloitre_, Sep 25 2004: (Start)
%F A014335 a(n+1) = Sum_{i=0..n} A000045(i)*2^(i-1).
%F A014335 a(n) = (1/5)*(2^(n-1)*A000032(n) - 1). (End)
%F A014335 a(n) = 2*a(n-1) + 4*a(n-2) + 1, a(0)=0; a(1)=0. - _Zerinvary Lajos_, Dec 14 2008
%F A014335 G.f.: G(0)*x^2/(2*(1-x)^2), where G(k)= 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F A014335 a(n) = (A203579(n) - 2)/5. - _Vladimir Reshetnikov_, Oct 06 2016
%p A014335 a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]+1 od: seq(a[n], n=0..29); # _Zerinvary Lajos_, Dec 14 2008
%p A014335 # second Maple program:
%p A014335 a:= n-> (<<0|1|0>, <0|0|1>, <-4|2|3>>^n)[1,3]:
%p A014335 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 04 2016
%t A014335 LinearRecurrence[{3,2,-4}, {0,0,1}, 41] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)
%t A014335 Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 06 2016 *)
%o A014335 (Magma) [(2^n*Lucas(n)-2)/10: n in [0..40]]; // _G. C. Greubel_, Jan 06 2023
%o A014335 (SageMath) [(2^n*lucas_number2(n,1,-1) -2)/10 for n in range(41)] # _G. C. Greubel_, Jan 06 2023
%Y A014335 Cf. A000032, A000045, A014334, A081057, A203579.
%Y A014335 Cf. (partial sums of) A063727.
%Y A014335 Column k=2 of A346415.
%K A014335 nonn,easy
%O A014335 0,4
%A A014335 _N. J. A. Sloane_