A203579 Exponential (or binomial) convolution of A000032 (Lucas) with itself, divided by 2.
2, 2, 7, 17, 57, 177, 577, 1857, 6017, 19457, 62977, 203777, 659457, 2134017, 6905857, 22347777, 72318977, 234029057, 757334017, 2450784257, 7930904577, 25664946177, 83053510657, 268766806017, 869747654657, 2814562533377, 9108115685377, 29474481504257
Offset: 0
Examples
With A000032 = {2,1,3,4,7,...},
2*a(4) = 1*2*7 + 4*1*4 + 6*3*3 + 4*4*1 + 1*7*2 = 114.
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..1961
- Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
Programs
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Mathematica
Array[Sum[Binomial[#, k] LucasL[k] LucasL[# - k], {k, 0, #}]/2 &, 28, 0] (* Michael De Vlieger, Dec 28 2020 *)
Formula
a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n)/2, n>=0, with L(n)=A000032(n).
E.g.f.: (1/2)*(exp(phi*x)+exp(-(phi-1)*x))^2 =
exp(x)*(cosh(sqrt(5)*x)+1), with the golden section phi:=(1+sqrt(5))/2. (See the e.g.f. of A000032).
a(n) = 2^(n-1)*L(n) + 1.
a(n) = 5*A014335(n) + 2. - Vladimir Reshetnikov, Oct 06 2016