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Math-NumSeq-33
River stage zero No dependents
3 ++
/ Math::NumSeq::Fibbinary

NAME

Math::NumSeq::Fibbinary -- without consecutive 1 bits

SYNOPSIS

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

DESCRIPTION

The fibbinary numbers 0, 1, 2, 4, 5, 8, 9, 10, etc, being integers which have no adjacent 1 bits in binary.

    i    fibbinary, and in binary
   ---   ------------------------
    0        0           0
    1        1           1
    2        2          10
    3        4         100
    4        5         101
    5        8        1000
    6        9        1001
    7       10        1010
    8       16       10000
    9       17       10001

For example at i=4 fibbinary 5 the next fibbinary is not 6 or 7 because they have adjacent 1 bits (110 and 111), the next is 8 (100).

Since the two highest bits must be "10...", skipping any high "11...", there's effectively a block of 2^k values (though not all of them used) followed by a gap of 2^k values, etc.

Zeckendorf Base

The bits of the fibbinary numbers encode Fibonacci numbers used to represent i in Zeckendorf style Fibonacci base. In that system an integer i is a sum of Fibonacci numbers,

    i = F(k1) + F(k2) + ... F(kn)         k1 > k2 > ... > kn

Each k is chosen as the highest Fibonacci less than the remainder at that point. For example, taking F(0)=1, F(1)=2, etc,

    20 = 13+5+1 = F(5)+F(3)+F(0)

The k's are then assembled as 1 bits in binary to encode the representation,

    fibbinary(20) = 2^5 + 2^3 + 2^0 = 41

The spacing between Fibonacci numbers means that after subtracting F(k) the next cannot be F(k-1), only F(k-2) or less. For that reason there's no adjacent 1 bits in the fibbinary numbers.

The connection between no adjacent 1s and the Fibonacci sequence can be seen from the values that have a high bit 2^k. New values with that high bit not with high 2^(k-1) but only 2^(k-2), so effectively the new values are not the previous k-1 but the second previous k-2, the same way as the Fibonacci sequence adds not the previous term but the one before.

FUNCTIONS

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

$seq = Math::NumSeq::Fibbinary->new ()

Create and return a new sequence object.

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

Return the $i'th fibbinary number.

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

Return true if $value is a fibbinary number, which means that in binary it doesn't have any consecutive 1 bits.

FORMULAS

Next Value

For a given fibbinary number, the next number is simply +1 if the lowest bit is 2^2=4 or more. If the low bit is 2^1=2 or 2^0=1 then the run of low alternating ...101 or ...1010 must be cleared and the bit above set. For example 1001010 becomes 1010000. All cases can be handled quite easily with some bit twiddling

    filled = (value >> 1) | value
    mask = ((filled+1) ^ filled) >> 1
    next value = (value | mask) + 1

For example

    value  = 1001010
    filled = 1101111
    mask   =    1111
    next   = 1010000

"filled" means any ...01010 ending has the zeros filled in to ...01111, then those low ones can be extracted with +1 and XOR (the usual trick for getting low ones). +1 means the bit above the filled part is included ...11111, but a shift drops back to "mask" of just 01111. OR-ing and incrementing then clears those low bits and sets the next higher to make ...10000.

This works for any fibbinary value inputs, both "...10101" and "...1010" endings and also zeros "...0000" ending. In the zeros case the result is just a +1 for "...0001", including input=0 giving next=1.

Ith Value

The i'th fibbinary number can be calculated by the Zeckendorf breakdown described above. Reckoning the Fibonacci numbers as F(0)=1, F(1)=2, F(2)=3, F(3)=5, etc,

    find the biggest F(k) <= i
    subtract i -= F(k)
    fibbinary result += 2^k
    repeat until i=0

To find each F(k)<=i either just work downwards through the Fibonacci numbers, or perhaps alternatively the Fibonaccis grow as (phi^k)/sqrt(5) with phi=(sqrt(5)+1)/2 the golden ratio, so an F(k) could be found by a log base phi of i. Or taking log2 of i (the highest bit in i) might give 2 or 3 candidates for k. Log base phi is unlikely to be faster, but log 2 high bit might quickly go to a nearly-correct place in a table.

Predicate

Testing for a fibbinary value can be done by a shift and AND,

    is_fibbinary = ((value & (value >> 1)) == 0)

Any adjacent 1 bits overlap in the shift+AND and come through as non-zero.

In Perl & converts NV float to UV/IV integer. If a value in an NV mantissa is an integer but bigger than a UV then bits will be lost in an &. Conversion to Math::BigInt or similar is necessary to preserve the full value. Floats which are integers but bigger than an UV/IV might be of interest, or it might be thought any float means rounded-off and therefore inaccurate and not of interest. The current code has some experimental automatic BigInt conversion which works and which accepts BigFloat or BigRat integers too, but don't rely on this quite yet. (A BigInt input directly is fine of course.)

Value Below

If a number is not a fibbinary, eg. 11000110, then the next lower fibbinary can be had by finding the highest 11 pair and changing it and all the bits below to 101010...etc. There's nothing in the code here doing this though.

Offending 11 pairs can be picked out by an AND per the predicate above. After a shift and AND the highest 1 bit is from the highest 11 pair and is the lower of the two bits in that pair. That position should then become a 0 etc, ie. "01010...". The higher 1 in the highest 11 pair is unchanged.

SEE ALSO

Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::FibonacciWord

HOME PAGE

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

LICENSE

Copyright 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.

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/>.