OFFSET
0,1
COMMENTS
The definition declares the Lucas numbers for all integers n and k. It gives the classical Lucas numbers as L(0, n) = Lucas(n), where Lucas(n) = A000032(n) is extended in the usual way for negative n.
LINKS
Peter Luschny, The Fibonacci Function.
FORMULA
Functional equation extends Cassini's theorem:
L(n, k) = (-1)^k*L(1 - n, -k - 2).
L(n, k) = n*Lucas(k + 1) + Lucas(k).
L(n, k) = L(n, k-1) + L(n, k-2).
L(n, k) = i^k*sec(c)*(n*cos(c*(k + 1)) - i*cos(c*k)), where c = Pi/2 + i*arccsch(2), for all n, k in Z.
Using the generalized Fibonacci numbers F(n, k) = A352744(n, k),
L(n, k) = F(n, k+1) + F(n, k) + F(n, k-1) + F(n, k-2).
EXAMPLE
Array starts:
[0] 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... A000032
[1] 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... A000032 (shifted)
[2] 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ... A000032 (shifted)
[3] 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, ... A022088
[4] 6, 13, 19, 32, 51, 83, 134, 217, 351, 568, ... A022388
[5] 7, 16, 23, 39, 62, 101, 163, 264, 427, 691, ... A190995
[6] 8, 19, 27, 46, 73, 119, 192, 311, 503, 814, ... A206420
[7] 9, 22, 31, 53, 84, 137, 221, 358, 579, 937, ... A206609
[8] 10, 25, 35, 60, 95, 155, 250, 405, 655, 1060, ...
[9] 11, 28, 39, 67, 106, 173, 279, 452, 731, 1183, ...
MAPLE
phi := (1 + sqrt(5))/2: psi := (1 - sqrt(5))/2:
L := (n, k) -> phi^(k+1)*(n - psi) + psi^(k+1)*(n - phi):
seq(seq(simplify(L(n-k, k)), k = 0..n), n = 0..10);
MATHEMATICA
L[n_, k_] := With[{c = Pi/2 + I*ArcCsch[2]},
I^k Sec[c] (n Cos[c (k + 1)] - I Cos[c k]) ];
Table[Simplify[L[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm
(* Alternative: *)
L[n_, k_] := n*LucasL[k + 1] + LucasL[k];
Table[Simplify[L[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm
PROG
(Julia)
const FibMem = Dict{Int, Tuple{BigInt, BigInt}}()
function FibRec(n::Int)
get!(FibMem, n) do
n == 0 && return (BigInt(0), BigInt(1))
a, b = FibRec(div(n, 2))
c = a * (b * 2 - a)
d = a * a + b * b
iseven(n) ? (c, d) : (d, c + d)
end
end
function Lucas(n, k)
k == 0 && return BigInt(n + 2)
k == -1 && return BigInt(2 * n - 1)
k < 0 && return (-1)^k * Lucas(1 - n, -k - 2)
a, b = FibRec(k)
c, d = FibRec(k - 1)
n * (2 * a + b) + 2 * c + d
end
for n in -6:6
println([Lucas(n, k) for k in -6:6])
end
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 29 2022
STATUS
approved