%I #49 Jan 06 2023 02:07:17

%S 0,0,1,3,11,35,115,371,1203,3891,12595,40755,131891,426803,1381171,

%T 4469555,14463795,46805811,151466803,490156851,1586180915,5132989235,

%U 16610702131,53753361203,173949530931,562912506675,1821623137075,5894896300851,19076285150003

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

%C 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 Alois P. Heinz, <a href="/A014335/b014335.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 (3,2,-4).

%F a(n) = A014334(n)/2.

%F G.f.: x^2/((1-x)*(1-2*x-4*x^2)). - _Vladeta Jovovic_, Mar 05 2003

%F E.g.f.: exp(x)*(cosh(sqrt(5)*x)-1)/5. - _Vladeta Jovovic_, Sep 01 2004

%F From _Benoit Cloitre_, Sep 25 2004: (Start)

%F a(n+1) = Sum_{i=0..n} A000045(i)*2^(i-1).

%F a(n) = (1/5)*(2^(n-1)*A000032(n) - 1). (End)

%F a(n) = 2*a(n-1) + 4*a(n-2) + 1, a(0)=0; a(1)=0. - _Zerinvary Lajos_, Dec 14 2008

%F 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 a(n) = (A203579(n) - 2)/5. - _Vladimir Reshetnikov_, Oct 06 2016

%p 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 # second Maple program:

%p a:= n-> (<<0|1|0>, <0|0|1>, <-4|2|3>>^n)[1,3]:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 04 2016

%t LinearRecurrence[{3,2,-4}, {0,0,1}, 41] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)

%t Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 06 2016 *)

%o (Magma) [(2^n*Lucas(n)-2)/10: n in [0..40]]; // _G. C. Greubel_, Jan 06 2023

%o (SageMath) [(2^n*lucas_number2(n,1,-1) -2)/10 for n in range(41)] # _G. C. Greubel_, Jan 06 2023

%Y Cf. A000032, A000045, A014334, A081057, A203579.

%Y Cf. (partial sums of) A063727.

%Y Column k=2 of A346415.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_