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A203579
Exponential (or binomial) convolution of A000032 (Lucas) with itself, divided by 2.
4
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
0,1
LINKS
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
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
EXAMPLE
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.
MATHEMATICA
Array[Sum[Binomial[#, k] LucasL[k] LucasL[# - k], {k, 0, #}]/2 &, 28, 0] (* Michael De Vlieger, Dec 28 2020 *)
CROSSREFS
Sequence in context: A051769 A256400 A203176 * A338415 A243022 A049955
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 14 2012
STATUS
approved