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

0,1

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.

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

Cf. A000032, A014335.

Sequence in context: A051769 A256400 A203176 * A338415 A243022 A049955

Adjacent sequences: A203576 A203577 A203578 * A203580 A203581 A203582

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 14 2012

Status

approved