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 A014335 Exponential convolution of Fibonacci numbers with themselves (divided by 2). 12
 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-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). 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 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 *) PROG (Magma) [(2^n*Lucas(n)-2)/10: n in [0..40]]; // G. C. Greubel, Jan 06 2023 (SageMath) [(2^n*lucas_number2(n, 1, -1) -2)/10 for n in range(41)] # G. C. Greubel, Jan 06 2023 CROSSREFS Cf. A000032, A000045, A014334, A081057, A203579. Cf. (partial sums of) A063727. Column k=2 of A346415. 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 June 1 08:50 EDT 2023. Contains 363068 sequences. (Running on oeis4.)