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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 1,3
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364,...
This is the same recurrence as the Fibonacci numbers (Math::NumSeq::Fibonacci), but a different starting point.
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::LucasNumbers->new ()
Create and return a new sequence object.
($i, $value) = $seq->next()
Return the next index and value in the sequence.
$valueexceeds the range of a Perl unsigned integer the return is a
Math::BigIntto preserve precision.
$value = $seq->ith($i)
$i'th Lucas number.
$bool = $seq->pred($value)
Return true if
$valueis 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.
Fibonacci F[k] and Lucas L[k] can be calculated together by a powering algorithm with two squares per doubling,
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
At the last step, ie. the lowest bit of i, only L[2k] or L[2k+1] needs to be calculated for the return, not the F too.
For any trailing zero bits of i, final doublings L[2k] can also be done with just one square as
L[2k] = L[k]^2 - 2*(-1)^k
The main double/step formulas can be applied until the lowest 1-bit of i is reached, then the L[2k+1] formula for that bit, followed by the single squaring for any trailing 0-bits.
L[i] increases as a power of phi, the golden ratio,
L[i] = phi^i + beta^i # exactly
So taking a log (natural logarithm) to get i, and ignoring beta^i which quickly becomes small,
log(L[i]) ~= i*log(phi) i ~= log(L[i]) / 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]) / 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 be close enough to take Lestimate = Festimate-1 (or vice-versa) and share code between the two.
Copyright 2010, 2011, 2012 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.
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