A354265 Array read by ascending antidiagonals for n >= 0 and k >= 0. Generalized Lucas numbers, L(n, k) = (psi^(k - 1)*(phi + n) - phi^(k - 1)*(psi + n)), where phi = (1 + sqrt(5))/2 and psi = (1 - sqrt(5))/2.

2, 3, 1, 4, 4, 3, 5, 7, 7, 4, 6, 10, 11, 11, 7, 7, 13, 15, 18, 18, 11, 8, 16, 19, 25, 29, 29, 18, 9, 19, 23, 32, 40, 47, 47, 29, 10, 22, 27, 39, 51, 65, 76, 76, 47, 11, 25, 31, 46, 62, 83, 105, 123, 123, 76, 12, 28, 35, 53, 73, 101, 134, 170, 199, 199, 123

Offset: 0

Author

Peter Luschny, May 29 2022

Keywords

• nonn
• tabl

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.

Examples

			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, ...
		

Links

• Peter Luschny, The Fibonacci Function.

Crossrefs

Cf. A000032, A000204, A022088, A022388, A190995, A206420, A206609.

Cf. A352744.

Programs

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

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

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).