Math::NumSeq::LucasNumbers -- Lucas numbers


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


The Lucas numbers, L(i) = L(i-1) + L(i-2) starting from values 1,3.

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

This is the same recurrence as the Fibonacci numbers (Math::NumSeq::Fibonacci), but a different starting point.

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

Each Lucas number falls in between successive Fibonaccis, and in fact the distance is a further Fibonacci,

    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]


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

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


See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::LucasNumbers->new ()
$seq = Math::NumSeq::LucasNumbers->new (i_start => $i)

Create and return a new sequence object.


($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.


Move the current sequence position to $i. The next call to next() will return $i and corresponding value.

Random Access

$value = $seq->ith($i)

Return the $i'th Lucas number.

$bool = $seq->pred($value)

Return true if $value is a Lucas number.

$i = $seq->value_to_i_estimate($value)

Return an estimate of the i corresponding to $value. See "Value to i Estimate" below.


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. This means add or subtract according to the previous bit handled.


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

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.

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


Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::Pell



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

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