A206609 -id:A206609 - cp's OEIS frontend

A206610 Fibonacci sequence beginning 13, 8.

13, 8, 21, 29, 50, 79, 129, 208, 337, 545, 882, 1427, 2309, 3736, 6045, 9781, 15826, 25607, 41433, 67040, 108473, 175513, 283986, 459499, 743485, 1202984, 1946469, 3149453, 5095922, 8245375, 13341297, 21586672, 34927969, 56514641, 91442610, 147957251

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, The Geometry of some Fibonacci Identities in the Hosoya Triangle, arXiv:1804.02481 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A206608, A206609.

I:=[13, 8]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012

Mathematica
```
LinearRecurrence[{1, 1}, {13, 8}, 80]
```

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.

Original entry on oeis.org

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

Peter Luschny, May 29 2022

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.

			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, ...

Peter Luschny, The Fibonacci Function.

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

Cf. A352744.

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

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);

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

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).

A206611 Fibonacci sequence beginning 13, 7.

Original entry on oeis.org

13, 7, 20, 27, 47, 74, 121, 195, 316, 511, 827, 1338, 2165, 3503, 5668, 9171, 14839, 24010, 38849, 62859, 101708, 164567, 266275, 430842, 697117, 1127959, 1825076, 2953035, 4778111, 7731146, 12509257, 20240403, 32749660, 52990063, 85739723, 138729786

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1).

Cf. A000032, A000045, A206609, A206610.

I:=[13, 7]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012

Mathematica
```
LinearRecurrence[{1, 1}, {13, 7}, 80]
```

cp's OEIS Frontend

A206610 Fibonacci sequence beginning 13, 8.

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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.

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A206611 Fibonacci sequence beginning 13, 7.

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Author

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Magma

Mathematica

cp's OEIS Frontend

A206610 Fibonacci sequence beginning 13, 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

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.

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Comments

Examples

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Programs

Julia

Maple

Mathematica

Formula

A206611 Fibonacci sequence beginning 13, 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

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.