A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)*(phi + n) - phi^(k - 1)*(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

0, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 3, 3, 3, 5, 1, 4, 4, 5, 5, 6, 1, 5, 5, 7, 8, 8, 7, 1, 6, 6, 9, 11, 13, 13, 8, 1, 7, 7, 11, 14, 18, 21, 21, 9, 1, 8, 8, 13, 17, 23, 29, 34, 34, 10, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 11, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89

Offset: 0

Author

Peter Luschny, May 09 2022

Keywords

• nonn
• tabl

Comments

The definition declares the Fibonacci numbers for all integers n and k. It gives the classical Fibonacci numbers as F(0, n) = A000045(n). A different enumeration is given in A352744.

Examples

			Array starts:
n\k 0, 1,  2,  3,  4,  5,  6,   7,   8,   9, ...
--------------------------------------------------------
[0]  0, 1,  1,  2,  3,  5,  8, 13,  21,  34, ... A000045
[1]  1, 1,  2,  3,  5,  8, 13, 21,  34,  55, ... A000045 (shifted once)
[2]  2, 1,  3,  4,  7, 11, 18, 29,  47,  76, ... A000032
[3]  3, 1,  4,  5,  9, 14, 23, 37,  60,  97, ... A104449
[4]  4, 1,  5,  6, 11, 17, 28, 45,  73, 118, ... [4] + A022095
[5]  5, 1,  6,  7, 13, 20, 33, 53,  86, 139, ... [5] + A022096
[6]  6, 1,  7,  8, 15, 23, 38, 61,  99, 160, ... [6] + A022097
[7]  7, 1,  8,  9, 17, 26, 43, 69, 112, 181, ... [7] + A022098
[8]  8, 1,  9, 10, 19, 29, 48, 77, 125, 202, ... [8] + A022099
[9]  9, 1, 10, 11, 21, 32, 53, 85, 138, 223, ... [9] + A022100
		

Links

• Peter Luschny, The Fibonacci Function.
• Wikipedia, Cassini and Catalan identities.

Crossrefs

Cf. A000045, A000032, A104449, A094588 (main diagonal).

Cf. A352744, A354265 (generalized Lucas numbers).

Programs

Julia

function fibrec(n::Int)
    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
function Fibonacci(n::Int, k::Int)
    k == 0 && return BigInt(n)
    k == 1 && return BigInt(1)
    k  < 0 && return (-1)^(k-1)*Fibonacci(-n - 1, 2 - k)
    a, b = fibrec(k - 1)
    a*n + b
end
for n in -6:6
    println([n], [Fibonacci(n, k) for k in -6:6])
end

Maple

f := n -> combinat:-fibonacci(n): F := (n, k) -> n*f(k - 1) + f(k):
seq(seq(F(n - k, k), k = 0..n), n = 0..11);
# The next implementation is for illustration only but is not recommended
# as it relies on floating point arithmetic. Illustrates the case n,k < 0.
phi := (1 + sqrt(5))/2: psi := (1 - sqrt(5))/2:
F := (n, k) -> (psi^(k-1)*(psi + n) - phi^(k-1)*(phi + n)) / (psi - phi):
for n from -6 to 6 do lprint(seq(simplify(F(n, k)), k = -6..6)) od;

Mathematica

(* Works also for n < 0 and k < 0. Uses a remark from Bill Gosper. *)
c := I*ArcSinh[1/2] - Pi/2;
F[n_, k_] := (n Sin[(k - 1) c] - I Sin[k c]) / (I^k Sqrt[5/4]);
Table[Simplify[F[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm

Formula

Functional equation extends Cassini's theorem:

F(n, k) = (-1)^(k - 1)*F(-n - 1, 2 - k).

F(n, k) = ((1 - phi)^(k - 1)*(1 - phi + n) - phi^(k - 1)*(phi + n))/(1 - 2*phi).

F(n, k) = n*fib(k - 1) + fib(k), where fib(n) are the classical Fibonacci numbers A000045 extended in the usual way for negative n.

F(n, k) - F(n-1, k) = fib(k-1).

F(n, k) = F(n, k-1) + F(n, k-2).

F(n, k) = (n*sin((k - 1)*c) - i*sin(k*c))/(i^k*sqrt(5/4)) where c = i*arcsinh(1/2) - Pi/2, for all n, k in Z. Based on a remark of Bill Gosper.