Math::N...Numbers(3)

💖 Math::NumSeq::LucasNumbers — the Lucas numbers bestie 💖

okay so this is the manual page for Math::NumSeq::LucasNumbers, from the User Contributed Perl Documentation, and it's genuinely one of my faves so let's GO 🦋

---

✨ NAME ✨

Math::NumSeq::LucasNumbers — Lucas numbers, periodt 💅

---

🌸 SYNOPSIS 🌸

use Math::NumSeq::LucasNumbers;
my $seq = Math::NumSeq::LucasNumbers->new;
my ($i, $value) = $seq->next;

that's it. that's the whole vibe check. 🫶

---

🎀 DESCRIPTION 🎀

The Lucas numbers!! They go L(i) = L(i-1) + L(i-2), starting from the values 1, 3. Like:

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364,...
starting i=1

It's literally the exact same recurrence as the Fibonacci numbers (that's Math::NumSeq::Fibonacci, iykyk), just a different starting point. Same energy, different origin story. main character energy tbh 👑

L[i+1] = L[i] + L[i-1]

And the way each Lucas number just falls in between successive Fibonaccis?? iconic. And the distance is a further Fibonacci, no cap:

F[i+1] < L[i] < F[i+2]

L[i] = F[i+1] + F[i-1]      # above F[i+1]
L[i] = F[i+2] - F[i-2]      # below F[i+2]

so it's squeezed in there, chillin' between two Fibonaccis, always. so cute 🥺

⭐ Start ⭐

Optional i_start => $i lets you start the sequence from somewhere other than the default i=1. For example, starting at i=0 gives value 2 at i=0, like:

i_start => 0
2, 1, 3, 4, 7, 11, 18, ...

you can just... move the whole thing. slay 💅

---

🦋 FUNCTIONS 🦋

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes. (that's the group chat lore everyone shares)

• $seq = Math::NumSeq::LucasNumbers->new ()
• $seq = Math::NumSeq::LucasNumbers->new (i_start => $i) — Create and return a new sequence object. new bestie acquired ✨

🌈 Iterating 🌈

• ($i, $value) = $seq->next() — Return the next index and value in the sequence.

When $value exceeds the range of a Perl unsigned integer the return is a Math::BigInt to preserve precision. no precision lost, we don't do that here 💀

• $seq->seek_to_i($i) — Move the current sequence position to $i. The next call to next() will return $i and corresponding value. like teleporting but for sequence indices 🧚

🍓 Random Access 🍓

• $value = $seq->ith($i) — Return the $i'th Lucas number. just ask for it directly, no waiting.
• $bool = $seq->pred($value) — Return true if $value is a Lucas number. is this a Lucas?? let's find out 🔥
• $i = $seq->value_to_i_estimate($value) — Return an estimate of the i corresponding to $value. See "Value to i Estimate" below. (it's an estimate, bestie, read the formula section for the deets)

---

👑 FORMULAS 👑

💜 Ith Pair 💜

A pair of Lucas numbers L[k], L[k+1] can be calculated together by a powering algorithm with two squares per doubling,

L[2k]   = L[k]^2   - 2*(-1)^k
L[2k+2] = L[k+1]^2 + 2*(-1)^k

L[2k+1] = L[2k+2] - L[2k]

if bit==0 then take L[2k], L[2k+1]
if bit==1 then take L[2k+1], L[2k+2]

The 2*(-1)^k terms mean adding or subtracting 2 according to k odd or even. Which means: add or subtract according to the previous bit handled. tracking bits like it's a situationship timeline 😭

🎀 Ith 🎀

For any trailing zero bits of i the final doublings can be done by L[2k] alone, which is one square per 0-bit.

ith_pair(odd part of i) to get L[2k+1]  (ignore L[2k+2])
loop L[2k] = L[k]^2 - 2*(-1)^k for each trailing 0-bit of i

efficiency?? we love to see it 💅

🌸 Value to i Estimate 🌸

L[i] increases as a power of phi, the golden ratio. The exact value is

L[i] = phi^i + beta^i    # exactly

phi = (1+sqrt(5))/2 = 1.618
beta = -1/phi = -0.618

Since abs(beta)<1 the beta^i term quickly becomes small. So taking a log (natural logarithm) to get i,

log(L[i]) ~= i*log(phi)
i ~= log(L[i]) * 1/log(phi)

Or the same using log base 2, which can be estimated from the highest bit position of a bignum,

log2(L[i]) ~= i*log2(phi)
i ~= log2(L[i]) * 1/log2(phi)

This is very close to the Fibonacci formula (see "Value to i Estimate" in Math::NumSeq::Fibonacci), being bigger by

Lestimate(value) - Festimate(value)
  = log(value) / log(phi) - (log(value) + log(phi-beta)) / log(phi)
  = -log(phi-beta) / log(phi)
  = -1.67

On that basis it could even be close enough to take Lestimate = Festimate-1, or vice-versa. they're basically the same girl, different font 🙈

🔥 Ith Fibonacci and Lucas 🔥

It's also possible to calculate a Fibonacci F[k] and Lucas L[k] together by a similar powering algorithm with two squares per doubling, but a couple of extra factors,

F[2k] = (F[k]+L[k])^2/2 - 3*F[k]^2 - 2*(-1)^k
L[2k] =                   5*F[k]^2 + 2*(-1)^k

F[2k+1] =    ((F[k]+L[k])/2)^2 + F[k]^2
L[2k+1] = 5*(((F[k]+L[k])/2)^2 - F[k]^2) - 4*(-1)^k

Or the conversions between a pair of Fibonacci and Lucas are

F[k]   = ( - L[k] + 2*L[k+1])/5
F[k+1] = ( 2*L[k] +   L[k+1])/5   = (F[k] + L[k])/2

L[k]   =  - F[k] + 2*F[k+1]
L[k+1] =  2*F[k] +   F[k+1]       = (5*F[k] + L[k])/2

two sequences, one glow-up, fr fr ✨

---

💫 SEE ALSO 💫

• Math::NumSeq
• Math::NumSeq::Fibonacci
• Math::NumSeq::Pell

the whole friend group 🫶

---

🌐 HOME PAGE 🌐

<http://user42.tuxfamily.org/math-numseq/index.html>

---

📜 LICENSE 📜

Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.

okay bestie real talk — the no-warranty bit is not a joke, it's doing actual legal work. respect the license 💜

---

and that's the whole page!! 🎀 this was perl v5.42.0, dated 2025-07-25, and it's the Math::N...Numbers(3) page — go forth and compute Lucas numbers, you absolute icon 👑💅✨