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A014335 Exponential convolution of Fibonacci numbers with themselves (divided by 2). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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-Fran├žois Alcover, Jun 15 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

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

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

a(n+1) = Sum_{i=0..n} F(i)*2^(i-1); a(n) = (1/5)*(2^(n-1)*L(n)-1) where L(n) are Lucas numbers defined in A000032. - Benoit Cloitre, Sep 25 2004

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

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

Join[{a=0, b=0}, Table[c=2*b+4*a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)

CROSSREFS

Cf. (partial sums of) A063727, A081057, A014334.

Sequence in context: A107683 A259400 A320087 * A147474 A247417 A222286

Adjacent sequences:  A014332 A014333 A014334 * A014336 A014337 A014338

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 24 11:49 EDT 2019. Contains 321448 sequences. (Running on oeis4.)