A070825 One half of product of first n+1 Lucas numbers A000032.

1, 1, 3, 12, 84, 924, 16632, 482328, 22669416, 1722875616, 211913700768, 42170826452832, 13579006117811904, 7074662187380001984, 5963940223961341672512, 8134814465483270041306368, 17953535525321576981163154176, 64112075360923351399733623562496, 370439571435415124387660876944101888

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..90

R. Grünwald, E. Heidel, A. Strätz, M. Sünkel and R. Terbach, Induction on Number Series, Fakultät fur Wirtschaftsinformatik und Angewandte Informatik, Otto-Friedrich-Universität Bamberg, 2012. - N. J. A. Sloane, Feb 07 2013

Formula

a(n) = (Product_{k=0..n} L(k))/2 with L = A000032.

Sum_{n>=0} 1/a(n) = 1 + A101690. - Amiram Eldar, Nov 09 2020

a(n) = 2^n*Product_{j=1..n} i^j*cosh(c*j), where c = arccsch(2) - i*Pi/2. - Peter Luschny, Jul 07 2025

Maple

c := arccsch(2) - I*Pi/2:
A070825 := n -> local j; 2^n*mul(I^j*cosh(c*j), j = 1..n):
seq(simplify(A070825(n)), n = 0..18);  # Peter Luschny, Jul 07 2025

Mathematica

FoldList[Times, LucasL[Range[0, 20]]]/2 (* or *)
Table[Round[GoldenRatio^(n(n+1)/2) QPochhammer[-1, GoldenRatio-2, n+1]]/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

Prog

(PARI) a(n) = prod(k=0, n, fibonacci(k+1)+fibonacci(k-1))/2; \\ Michel Marcus, Mar 18 2016
(Magma) [1] cat [&*[Lucas(i+1): i in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 15 2016

Crossrefs

Cf. A000032, A003266 (for Fibonacci), A003046 (for Catalan), A101690, A135407, A218490.

Sequence in context: A032183 A225905 A355322 * A232934 A077047 A074505

Adjacent sequences: A070822 A070823 A070824 * A070826 A070827 A070828

Keyword

nonn,easy

Author

Wolfdieter Lang, May 10 2002

Status

approved