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

0, 0, 1, 3, 11, 35, 115, 371, 1203, 3891, 12595, 40755, 131891, 426803, 1381171, 4469555, 14463795, 46805811, 151466803, 490156851, 1586180915, 5132989235, 16610702131, 53753361203, 173949530931, 562912506675, 1821623137075, 5894896300851, 19076285150003

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Crossrefs

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

Cf. (partial sums of) A063727.

Column k=2 of A346415.

Programs

Magma

[(2^n*Lucas(n)-2)/10: n in [0..40]]; // G. C. Greubel, Jan 06 2023

Maple

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
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <-4|2|3>>^n)[1,3]:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 04 2016

Mathematica

LinearRecurrence[{3,2,-4}, {0,0,1}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)

SageMath

[(2^n*lucas_number2(n,1,-1) -2)/10 for n in range(41)] # G. C. Greubel, Jan 06 2023

Formula

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

G.f.: x^2/((1-x)*(1-2*x-4*x^2)). - Vladeta Jovovic, Mar 05 2003

E.g.f.: exp(x)*(cosh(sqrt(5)*x)-1)/5. - Vladeta Jovovic, Sep 01 2004

From Benoit Cloitre, Sep 25 2004: (Start)

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

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

a(n) = 2*a(n-1) + 4*a(n-2) + 1, a(0)=0; a(1)=0. - Zerinvary Lajos, Dec 14 2008

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

a(n) = (A203579(n) - 2)/5. - Vladimir Reshetnikov, Oct 06 2016