++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::NumSeq::GolayRudinShapiroCumulative -- cumulative Golay/RudinShapiro sequence

# SYNOPSIS

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

# DESCRIPTION

This is the Golay/Rudin/Shapiro sequence values accumulated as GRS(0)+...+GRS(i),

starting from i=0 value=GRS(0)

1, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 4, ...

The total is always positive, and in fact a given cumulative total k occurs precisely k times. For example the three occurrences of 3 shown above are all the places 3 occurs.

This GRS cumulative arises as in the alternate paper folding curve as the coordinate sum X+Y. The way k occurs k many times has a geometric interpretation as the points on the diagonal X+Y=k of the curve visited a total of k many times. See "dSum" in Math::PlanePath::AlternatePaper.

# FUNCTIONS

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

\$seq = Math::NumSeq::GolayRudinShapiroCumulative->new ()

Create and return a new sequence object.

## Random Access

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

Return the \$i'th value from the sequence, being the total GRS(0)+GRS(1)+...+GRS(\$i).

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

Return true if \$value occurs in the sequence. All positive integers occur, so this simply means integer \$value >= 1.

# FORMULAS

## Ith

The cumulative total GRS(0)+...+GRS(i-1) can be calculated from the 1-bits of i. Each 1-bit becomes a value 2^floor((pos+1)/2) in the total,

bit    value
---    -----
0       1
1       2
2       2
3       4
4       4
...     ...
k      2^ceil(k/2)

The value is added or subtracted from the total according to the number of 11 bit pairs above that bit position, not including the bit itself,

sub value    if odd count

For example i=27 is 110011 in binary so

1      -1      bit0 low bit
1      -2      bit1
0              bit2
1      +4      bit3
1      +4      bit4 high bit
----
5      cumulative value GRS(0)+...+GRS(26)

The second lowest bit is negated as value -2 because there's one "11" bit pair above it, and -1 the same because above and not including that bit there's just one "11" bit pair.

Or for example i=31 is 11111 in binary so

1      -1      bit0 low bit
1      +2      bit1
1      -2      bit2
1      +4      bit3
1      +4      bit4 high bit
----
7      cumulative total GRS(0)+...+GRS(30)

Here at bit2 the value is -2 because there's one adjacent 11 above, not including bit2 itself. Then at bit1 there's two 11 pairs above so +2, and at bit0 there's three so -1.

The total can be formed by examining the bits high to low and counting adjacent 11 bits on the way down to add or subtract. Or it can be formed from low to high by negating the total so far when a 11 pair is encountered.

For an inclusive sum GRS(0)+...+GRS(i) as per this module, the extra GRS(i) can be worked into the calculation by its GRS definition +1 or -1 according to the total number of adjacent 11 bits. This can be thought of as an extra value 1 below the least significant bit. For example i=27 inclusive

+1      below all bits
1      -1      bit0 low bit
1      -2      bit1
0              bit2
1      +4      bit3
1      +4      bit4 high bit
----
5      cumulative value GRS(0)+...+GRS(27)

For low to high calculation this lowest +/-1 can be handled simply by starting the total at 1. It then becomes +1 or -1 by the negations as 11s are encountered for the rest of the bit handling.

total = 1   # initial value below all bits to be inclusive GRS(i)
power = 1   # 2^ceil(bitpos/2)
thisbit = take bit from low end of i

loop
nextbit = take bit from low end of i
if thisbit&&nextbit
then total = -total    # negate lower values added
if thisbit
then total += power
thisbit = nextbit

power *= 2
exit loop if i==0

nextbit = bit from low end of i
if thisbit&&nextbit
then total = -total    # negate lower values added
if thisbit
then total += power
thisbit = nextbit
exit loop if i==0
endloop

total += power     # final for highest 1-bit in i
# total=GRS(0)+...+GRS(i)

This sort of calculation arises implicitly in the alternate paper folding curve to calculate X,Y for a given N point on the curve. But that calculation does a simultaneous using the base 4 digits of N.

X=GRStotal(ceil(N/2))
Y=GRStotal(floor(N/2))

Math::PlanePath::AlternatePaper