a binary sequence generator based on the kolakoski sequence and multiples of odd primes

Page 1

KA

MA

WO

TA

N v

ol.

1,

no. 2

, M

ay

20

07

78

Sec

on

dary

So

urce

s

Ber

nasc

oni,

R.

(198

7).

De

cons

truc

tion

and

the

pos

sibi

lity

of e

thic

s. I

n J.

S

alli

s (E

d.),

Dec

ons

truc

tion

an

d p

hilo

sop

hy

. C

hica

go:

Uni

vers

ity o

f C

hica

go P

ress

.

Ca

puto

, J.

(19

93a

). Ag

ains

t e

thic

s: C

on

trib

utio

ns t

o a

po

etic

s of

ob

liga

tion

with

con

sta

nt r

efe

ren

ce t

o d

eco

nstr

uct

ion

, S

tudi

es i

n C

ontin

enta

l T

hou

ght.

Blo

omin

gton

: In

dia

na U

nive

rsity

Pre

ss.

Ca

puto

, J.

(19

93b)

. Ra

dica

l he

rme

neu

tics:

Re

petit

ion

, de

con

stru

ctio

n a

nd

the

her

me

neu

tic p

roje

ct, S

tudi

es in

Phe

nom

enol

ogy

and

Exi

sten

tial

Phi

loso

phy.

Blo

omin

gton

: In

dia

na U

nive

rsity

Pre

ss.

Ca

puto

, J.

(1

997a

). De

con

stru

ctio

n

in

a

nut

shel

l.

New

Y

ork:

F

ordh

am

Uni

vers

ity P

ress

.

Ca

puto

, J.

(19

97b)

. Th

e p

raye

rs a

nd

te

ars

of J

acq

ues

De

rrid

a:

Rel

igio

n w

itho

ut r

elig

ion.

Ind

iana

polis

: Ind

iana

Uni

vers

ity P

ress

.

Cor

nell,

D.

(Ed.

). (

1992

). De

cons

tru

ctio

n a

nd t

he

pos

sib

ility

of

just

ice

. N

ew

Yor

k: R

outle

dge.

Cri

tchl

ey,

S.

(199

0).

Der

rida

a

nd

deco

nstr

uctio

n.

In

R.

Kea

rne

y (E

d.),

R

ou

tledg

e h

isto

ry o

f p

hilo

sop

hy (vo

l. V

II).

Lond

on a

nd N

ew Y

ork:

R

outle

dge.

Cri

tchl

ey,

S.

(199

2). T

he e

thic

s of

de

con

stru

ctio

n: D

errid

a a

nd

Lev

ina

s.

Oxf

ord:

Bla

ckw

ell.

Cul

ler,

J. (

1983

). On

dec

ons

truc

tion. L

ondo

n: R

outle

dge

and

Keg

an

Pa

ul.

Kea

rne

y, R

. (19

99).

Po

etic

s of

mo

der

nity. N

ew Y

ork:

Hum

anity

Boo

ks.

Levi

nas,

E.

(196

9). T

ota

lity

and

inf

inity

. (A

. Li

ngis

, T

rans

.).

Pitt

sbur

gh:

Duq

uesn

e U

nive

rsity

Pre

ss.

Nor

ris, C

. (1

987)

. De

rrid

a. C

am

brid

ge:

Ha

rva

rd U

nive

rsity

Pre

ss.

Ra

papo

rt,

H. (

2003

). La

ter

Der

rida.

New

Yor

k: R

outle

dge.

Sim

, S. (

1992

). Bey

on

d a

esth

etic

s. Lon

don:

Ha

rves

ter

Whe

ats

hea

f.

Col

lege

of

Art

s a

nd

Sci

en

ces,

Ate

neo

de

Na

ga

Un

ive

rsi

ty

79

A B

inar

y S

eque

nce

Gen

erat

or B

ased

on

the

Kol

akos

ki S

eque

nce

and

Mu

ltip

les

of O

dd

Pri

mes

Joel

R. N

oche

D

epa

rtm

ent

of M

ath

em

atic

s a

nd N

atur

al S

cien

ces

A

bstr

act

An

ear

lier

bin

ary

seq

uen

ce g

ene

rato

r is

mo

difi

ed

to

yie

ld

be

tter

per

form

an

ce i

n t

he D

IEH

AR

D t

ests

for

ran

dom

ne

ss

of

un

iform

ly

dis

trib

ute

d

num

ber

s.

T

he

n

ew

g

ene

rato

r in

vert

s e

very

p

bits

of

a

mo

difi

ed

K

ola

kosk

i se

qu

ence

, w

he

re p

is a

prim

e n

um

ber

, a

nd d

oes

this

for

the

fi

rst f

od

d p

rim

e

num

ber

s.

S

amp

les

from

th

e

resu

ltin

g

bina

ry

sequ

enc

e c

an

be

use

d i

n s

cie

ntifi

c a

ppl

ica

tions

re

qui

rin

g ra

nd

om n

umbe

rs.

Intr

oduc

tion

Larg

e a

mo

unt

s of

“ra

ndo

m”

num

ber

s ar

e cu

rren

tly n

eed

ed

by

ma

ny

ap

plic

atio

ns (

Hay

es,

20

01)

. I

n th

is w

ork,

num

ber

s a

re s

aid

to

be

rand

om

if t

hey

perf

orm

wel

l in

com

mo

nly

acc

ept

ed

sta

tistic

al t

ests

of

ra

nd

om

nes

s.

Pse

ud

o-ra

ndo

m n

um

ber

ge

nera

tors

(P

RN

Gs)

usu

ally

re

fer

to g

ene

rato

rs o

f u

nifo

rmly

dis

trib

uted

nu

mb

ers

(oft

en o

ver

the

in

terv

al

(0,1

)).

(A

rbitr

ary

dist

rib

utio

ns

are

us

ually ge

nera

ted

by

app

lyin

g tr

ans

form

atio

ns t

o th

ese

valu

es.)

P

RN

Gs

diffe

r fr

om

(tr

ue)

rand

om

n

um

ber

g

ener

ato

rs

(RN

Gs)

in

th

at

the

form

er

per

form

de

term

inis

tic o

per

atio

ns o

n a

see

d va

lue,

whi

le t

he l

att

er r

ely

on a

n

on-d

eter

min

istic

so

urce

of

rand

om

nes

s (u

sua

lly a

phy

sica

l pr

oces

s su

ch

as

radi

oact

ive

deca

y or

th

erm

al

noi

se).

Tw

o (o

f m

any

) pr

ope

rtie

s th

at

a “

goo

d” P

RN

G o

r R

NG

sho

uld

have

are

: (

1)

the

num

ber

s of

bits

‘0’

and

‘1

’ it

prod

uces

sh

ould

be

alm

ost

eq

ual,

and

(2

) th

e se

que

nce

sho

uld

not

rep

eat,

tha

t is,

it s

houl

d b

e n

on-

peri

odic

.

In a

n ea

rlie

r w

ork

(Noc

he,

20

04)

, a

cer

tain

bin

ary

se

que

nce

ge

nera

tor

wa

s pr

opos

ed.

It

star

ted

with

a T

hu

e-M

orse

seq

uenc

e 0

, 1

, 1

, 0

, 1

, 0

, 0,

1,

1, 0

, 0

, 1,

0,

1, 1

, 0

, …

(se

que

nce

A01

00

60

in S

loa

ne

(2

006

)).

Thi

s se

que

nce

is n

ot p

erio

dic

(Allo

uch

e &

Sh

alli

t, 1

99

9) a

nd

the

nu

mb

ers

of ‘

0’s

and

‘1

’s i

n a

n in

finite

Th

ue-M

orse

se

que

nce

are

th

e sa

me.

T

his

seq

uenc

e, h

ow

eve

r, t

erri

bly

fa

ils c

om

mo

nly

acc

ept

ed

st

atis

tica

l tes

ts o

f ra

ndo

mn

ess.

It w

as s

how

n in

(N

och

e, 2

00

4) th

at

the

alg

orith

m u

sed

in t

hat

wor

k to

ge

nera

te t

he T

hue

-Mor

se s

eque

nce

lea

d to

an

int

eres

ting

Page 2

KA

MA

WO

TA

N v

ol.

1,

no. 2

, M

ay

20

07

80

met

hod

of

ad

din

g “d

isor

der”

to

the

orig

ina

l seq

ue

nce

. T

he b

its w

hos

e

orde

r in

th

e se

qu

enc

e w

as

a m

ultip

le o

f a

pri

me

nu

mb

er p

wer

e in

vert

ed,

an

d th

is w

as d

one

seq

uen

tially

for

incr

easi

ng

p fo

r th

e fir

st f

odd

pri

me

nu

mb

ers.

(O

dd p

rim

e n

um

ber

s w

ere

chos

en b

eca

use

no

“dis

orde

r” i

s in

trod

uce

d b

y in

vert

ing

all

the

eve

n b

its,

tha

t is

, th

e m

ultip

les

of t

he o

nly

eve

n pr

ime

nu

mb

er,

2.)

Thi

s ge

nera

tor

is

now

m

odifi

ed

by

sta

rtin

g it

with

a

K

ola

kosk

i seq

uenc

e in

stea

d.

A K

ola

kosk

i seq

uen

ce is

a

seq

uenc

e th

at

equa

ls it

s ow

n ru

n le

ngt

h se

que

nce

(K

ola

kosk

i, 1

96

5).

T

he s

eque

nce

()

{}∞ =0n

nk

= 1

, 2,

2,

1, 1

, 2,

1,

2, 2

, 1,

2,

2, 1

, 1,

2,

… (

sequ

en

ce

A0

00

002

in S

loa

ne (

20

06

)) is

the

Kol

ako

ski s

equ

enc

e t

hat

star

ts w

ith

‘1.’

Lis

ted

abo

ve a

re it

s fir

st t

en r

uns;

the

fir

st r

un is

‘1,’

the

seco

nd

run

is ‘2

,2,’

the

thir

d ru

n is

‘1,1

,’ th

e fo

urth

ru

n is

‘2,’

and

so

on.

T

he

fir

st r

un h

as k

(1)

= 1

ele

me

nt,

the

seco

nd

run

has

k(2)

= 2

ele

men

ts,

the

thir

d ru

n ha

s k(3

) =

2 e

lem

ent

s, t

he f

ourt

h ru

n h

as k(4)

= 1

el

em

ent

, a

nd s

o o

n.

The

Kol

ako

ski

seq

uenc

e th

at

star

ts w

ith ‘2

’ is

2,

2, 1

, 1

, 2

, 1

, 2

, 2

, 1

, 2

, 2,

1,

1, 2

, 1,

1,

… (

seq

uenc

e A

07

888

0 i

n S

loa

ne

(20

06))

.

Thi

s is

the

ver

sio

n a

rbitr

arily

cho

sen

in t

his

wor

k.

A b

ina

ry s

eque

nce

co

mp

ose

d of

‘0

’s

and

‘1

’s

is

desi

red,

so

the

‘2’

s a

re a

rbi

trar

ily

conv

erte

d in

to ‘0

’s t

o g

et th

e b

ina

ry s

eque

nce (

){

}∞ =0nn

x =

0,

0,

1, 1

, 0,

1,

0, 0

, 1

, 0,

0,

1, 1

, 0,

1,

1, …

.

The

K

ola

kosk

i se

que

nce

is

not

peri

odic

(K

ola

kosk

i a

nd

Uco

luk,

19

66

) a

nd i

t is

co

njec

ture

d (b

ut n

ot y

et p

ro

ven)

tha

t th

e

num

ber

of

‘0’s

and

‘1

’s i

n an

inf

inite

Kol

ako

ski

seq

uen

ce a

re t

he

sa

me

(Wei

sste

in, 2

00

6).

Met

hodo

logy

One

pop

ula

r w

ay

to t

est

PR

NG

s a

nd R

NG

s is

the

DIE

HA

RD

se

ries

of t

ests

(M

ars

agl

ia,

199

5).

DIE

HA

RD

co

ntai

ns fi

fte

en

test

s fo

r ra

ndo

mn

ess

of u

nifo

rmly

dis

trib

uted

nu

mb

ers.

C

are

sho

uld

be

take

n

whe

n ru

nni

ng

thes

e te

sts.

F

or e

xam

ple,

Mar

sagl

ia r

epor

ts t

hat

the

hard

wa

re

RN

Gs

he

test

ed

faile

d “s

pect

acu

larl

y.”

D

avi

es

(19

97)

fo

un

d th

at

the

sup

pose

d fa

ilure

s w

ere

mos

tly d

ue

to a

da

ta h

an

dlin

g

erro

r:

each

byt

e w

ith a

va

lue

of 1

0 w

as

bei

ng r

ecor

de

d a

s tw

o b

ytes

w

ith v

alu

es o

f 13

and

10

(a

car

ria

ge r

etur

n fo

llow

ed b

y a

lin

e fe

ed)

.

An

othe

r po

ssib

le p

rob

lem

is t

he d

iffer

ent

con

vent

ions

fo

r th

e or

der

in

whi

ch t

he

bits

or

byt

es a

re s

tore

d in

a w

ord.

T

he

rea

rra

nge

me

nt o

f b

ytes

whe

n c

op

yin

g b

etw

een

com

put

ers

with

diff

eren

t c

on

vent

ions

Col

lege

of

Art

s a

nd

Sci

en

ces,

Ate

neo

de

Na

ga

Un

ive

rsi

ty

81

can

affe

ct t

he r

esul

ts o

f co

rrel

atio

ns t

ests

an

d ot

her

test

s (D

avie

s,

200

0).

In

thi

s w

ork,

bot

h pr

oble

ms

are

avo

ide

d:

the

out

put

is

save

d

as

a b

inar

y fil

e (n

ot a

tex

t fil

e),

and

onl

y b

yte-

siz

ed w

ord

s ar

e us

ed

to

repr

esen

t th

e se

qu

ence

. D

eta

ils o

f th

e te

sts

an

d th

e

ass

ocia

ted p

-va

lues

are

in

Ma

rsa

glia

(1

99

5).

The

res

ults

of

the

test

s ca

n b

e

sum

ma

rize

d b

y 1

46

p-

valu

es.

The

pro

gra

m p

rese

nte

d in

(N

och

e, 2

00

4)

calle

d su

bro

utin

es

fro

m in

side

loo

ps.

Th

e re

vise

d ve

rsio

n lis

ted

in A

ppe

ndi

x A

doe

s n

ot

use

sub

rout

ine

calls

, res

ultin

g in

impr

ove

d p

erfo

rma

nce

.

Fo

ur c

ase

s a

re t

este

d:

thos

e in

volv

ing

the

first

f

= 1

0,

10

0,

100

0,

and

20

00

odd

pri

me

nu

mb

ers.

T

he

case

s la

bele

d t

10

, t1

00

, t1

00

0, a

nd t

20

00 u

se t

he T

hu

e-M

orse

seq

uenc

e, a

nd

tho

se l

ab

eled

k1

0,

k10

0,

k10

00

, a

nd k

20

00

use

the

Kol

ako

ski s

eque

nc

e.

(The

latt

er

four

ca

ses

use

the

pro

gra

m l

iste

d in

Ap

pen

dix

B.)

E

ach

seq

ue

nce

is

16,

77

7,2

16

byte

s lo

ng.

(D

IEH

AR

D n

eeds

at

lea

st a

n 8

0 m

illio

n b

it lo

ng te

st s

eque

nce

for

all

of it

s te

sts

to r

un.)

For

co

mpa

riso

n, t

he r

esul

ts o

f th

e P

RN

G K

ISS

(M

arsa

gli

a,

199

5)

are

also

inc

lud

ed.

It

“se

ems

to p

ass

all

[the

DIE

HA

RD

] te

sts

and

is

hig

hly

reco

mm

en

ded

for

sp

eed

and

sim

plic

ity”

(Mar

sagl

ia,

199

5).

T

he

KIS

S s

equ

enc

e te

ste

d h

ere

is 1

1,4

68,

80

0 b

yte

s lo

ng a

nd

use

d th

e se

ed

inte

gers

1,

4, 3

, a

nd 4

4.

Res

ults

Ta

ble

1 s

how

s th

e ru

n tim

es f

or a

co

mp

uter

with

an

Int

el

Cel

eron

1.7

0 G

Hz

proc

esso

r, 2

56

MB

of

RA

M,

and

a M

icr

osof

t W

ind

ows

XP

ope

ratin

g sy

ste

m.

The

KIS

S o

utpu

t fil

e ta

kes

less

tha

n

a s

econ

d to

ge

nera

te a

fter

the

seed

va

lues

are

giv

en.

Ta

ble

1. R

un ti

mes

(in

sec

onds

)

f T

hue-

Mor

se

(Noc

he,

20

04)

T

hue

-Mor

se

(App

endi

x A

) K

ola

kosk

i (A

ppen

dix

B)

10

1

3

4 8

1

00

18

7

11

1

00

0 2

6

15

1

8

20

00

29

1

6

20

The

p-v

alu

es f

or t

he t

10

, t1

00,

t1

000

, t2

00

0,

and

KIS

S t

est

ca

ses

are

lis

ted

in (

Noc

he,

20

04

).

(Th

e pr

ogr

am

in

Ap

pen

dix

A

yiel

ds t

he s

am

e D

IEH

AR

D

p-va

lues

as

the

pro

gra

m in

Noc

he

(20

04).

)

Page 3

KA

MA

WO

TA

N v

ol.

1,

no. 2

, M

ay

20

07

82

Due

to

spa

ce l

imita

tions

, th

e p-va

lues

for

the

k1

0,

k10

0,

k10

00

, an

d

k20

00

ca

ses

are

not

liste

d he

re.

Dis

cuss

ion

It i

s im

plie

d t

hat

if a

ll th

e p-

valu

es o

f a

seq

ue

nce

tes

ted

by

DIE

HA

RD

lie

in t

he r

ang

e (0

.025

, 0.

97

5) t

hen

the

seq

uen

ce is

sa

id t

o ha

ve p

ass

ed a

ll th

e D

IEH

AR

D t

ests

. T

his

is t

he r

easo

ning

be

hin

d th

e

cla

im in

(N

oche

, 20

04

) th

at

the

seq

uen

ce t

20

00

per

form

ed

wel

l in

the

te

sts.

A

sub

seq

uent

ana

lysi

s of

the

da

ta,

how

eve

r, s

how

s th

at

t20

00’

s te

st r

esul

ts s

till l

eave

mor

e to

be

desi

red.

The

his

togr

am

s of

the

p-va

lues

for

ea

ch T

hue

-Mor

se-b

ased

te

st c

ase

are

sho

wn

in F

igur

e 1

. N

ote

tha

t m

ost

of t

he p

-va

lues

are

in

the

upp

erm

ost

deci

le.

The

his

togr

am

for

the

KIS

S t

est

ca

se (

Fig

ure

2)

is m

ore

uni

form

ly d

istr

ibut

ed.

Thi

s se

em

s to

im

ply

tha

t a

“m

ore

ra

ndo

m”

seq

ue

nce

wo

uld

have

a m

ore

uni

form

ly d

istr

ibu

ted

set

of p-

valu

es in

the

DIE

HA

RD

test

s.

Fig

ure

1.

His

togr

am

s of

DIE

HA

RD

p-va

lues

(T

hue-

Mor

se-b

ase

d se

quen

ces)

0.0-

0.1

0.1

-0.2

0.2-

0.3

0.3-

0.4

0.4

-0.5

0.5

-0.6

0.6

-0.7

0.7

-0.8

0.8-

0.9

0.9

-1.0

t200

0

t100

0

t100t1

00

20406080

100

120

140

Col

lege

of

Art

s a

nd

Sci

en

ces,

Ate

neo

de

Na

ga

Un

ive

rsi

ty

83

Fig

ure

2.

His

togr

am

of D

IEH

AR

D p-

valu

es (

KIS

S s

eque

nce

)

0.0-

0.1

0.1-

0.2

0.2-

0.3

0.3-

0.4

0.4-

0.5

0.5-

0.6

0.6-

0.7

0.7-

0.8

0.8-

0.9

0.9-

1.0

KIS

S

0

20406080

100

120

140

Fig

ure

3.

His

togr

am

s of

DIE

HA

RD

p-va

lues

(K

ola

kosk

i-ba

sed

sequ

ence

s)

0.0-

0.1

0.1-

0.2

0.2

-0.3

0.3

-0.4

0.4-

0.5

0.5

-0.6

0.6-

0.7

0.7

-0.8

0.8-

0.9

0.9

-1.0

k200

0

k100

0

k10

0

k10

0

20406080

100

120

140

Page 4

KA

MA

WO

TA

N v

ol.

1,

no. 2

, M

ay

20

07

84

Fig

ure

3 s

how

s th

e hi

stog

ram

s fo

r th

e K

ola

kosk

i-ba

sed

test

ca

ses.

It

show

s th

at e

ven

case

k1

00

see

ms

to h

ave

be

tter

te

st r

esul

ts

tha

n ca

se t

20

00.

It m

ust

be

me

ntio

ned,

how

eve

r, t

hat

all

the

Kol

ako

ski

-bas

ed

test

ca

ses

perf

orm

ed

very

p

oor

ly

in

the

se

vent

h D

IEH

AR

D

test

(“

Cou

nt t

he 1

’s in

a s

trea

m o

f b

ytes

”).

Thi

s ca

n b

e s

een

in T

ab

le 2

.

Ta

ble

2. p

-va

lue

s fo

r th

e se

vent

h D

IEH

AR

D te

st (

roun

ded

off)

.

t1

0 t1

00

t100

0 t2

000

K

ISS

k1

0 k1

00

k100

0 k2

000

p 109

1

.000

0.

979

0.96

3 0.

997

0.9

19

1.00

0 1.

000

1.00

0 0.

993

p 110

1

.000

1.

000

1.00

0 0.

835

0.3

36

1.00

0 1.

000

1.00

0 1.

000

Con

clus

ions

The

bin

ary

seq

uenc

e ge

ner

ato

r pr

esen

ted

her

e is

sh

own

to

pe

rfor

m a

rgua

bly

bet

ter

in a

pop

ular

set

of

test

s fo

r r

and

om

nes

s th

an

a

n ea

rlie

r w

ork.

In

pa

rtic

ula

r, c

ase

k1

00 h

as

a s

hor

ter

run

tim

e th

an

ca

se t

20

00

and

its

p-

valu

es a

re m

ore

uni

form

ly d

istr

ibut

ed

tha

n th

ose

of

ca

se t

20

00.

Ref

eren

ces

Allo

uche

, J.

-P.,

& S

halli

t, J.

(19

99).

The

ubi

quitous

Pro

uhet

-Thu

e-M

orse

se

quen

ce.

In Se

que

nce

s a

nd T

heir

A

ppl

ica

tions

: P

roce

edin

gs of

S

ET

A ’9

8, S

prin

ger-

Ver

lag,

1–1

6.

Da

vies

, R

. (1

997)

. Tru

e r

an

dom

num

ber

ge

nera

tors. R

etri

eved

Apr

il 20

04,

from

http

://w

ww

.rob

ertn

z. n

et/tr

ue_m

g.ht

ml

Da

vies

, R

. (2

000)

. Ha

rdw

are

ran

dom

num

ber

gen

erat

ors

. R

etrie

ved

Apr

il 20

04, f

rom

http

://w

ww

.rob

ertn

z.ne

t/hw

rng.

htm

l

Ha

yes,

B.

(200

1).

Com

putin

g sc

ienc

e: R

ando

mne

ss a

s a

res

ourc

e, A

mer

ica

n S

cie

ntis

t, 89(

4), 3

00–3

04.

Kol

ako

ski,

W.

(196

5).

Sel

f ge

nera

ting

runs

, P

robl

em 53

04.

Am

eric

an

Ma

the

ma

tical

Mo

nth

ly, 7

2(6)

, 674

.

Kol

ako

ski,

W.,

&

Uco

luk,

N

. (1

966)

. 53

04. A

me

rican

M

ath

em

atic

al

Mo

nth

ly, 7

3(6)

, 68

1–68

2.

Mar

sagl

ia,

G.

(199

5). The

Mar

sag

lia r

an

dom

num

ber

CD

RO

M in

clu

din

g t

he

DIE

HA

RD

bat

tery

of

test

s of

ra

ndom

ness

. R

etrie

ved

Ma

rch

2004

, fr

om h

ttp://

sta

t.fsu

.edu

/pub

/die

hard

/

Noc

he,

J. (

2004

). A

bin

ary

seq

uenc

e ge

nera

tor

base

d o

n th

e T

hue-

Mor

se

sequ

ence

a

nd

mul

tiple

s of

pr

imes

. T

he

Ph

ilipp

ine

S

tatis

ticia

n, 5

3(1–

4),

67–7

6.

Col

lege

of

Art

s a

nd

Sci

en

ces,

Ate

neo

de

Na

ga

Un

ive

rsi

ty

85

Slo

ane

, N

. (2

006)

. Th

e o

n-li

ne e

ncyc

lope

dia

of

inte

ger

sequ

en

ces

. R

etri

eved

A

pril

2007

, fr

om h

ttp:

//ww

w.r

esea

rch.

att.c

om/~

njas

/se

quen

ces/

Wei

sste

in,

E.

(200

6). K

olak

oski

se

que

nce. F

rom

Ma

thW

orld

—A

Wo

lfra

m

We

b

Re

sour

ce. R

etrie

ved

Apr

il 20

07

from

ht

tp://

ma

thw

orld

.w

olfr

am

.com

/Kol

ako

skiS

eque

nce.

htm

l

App

endi

x A

/* improved version of the program presented in (Noche, 2004) */

/* source code for Microsoft Visual C++ 6.0 */

#include <stdio.h>

#include <stdlib.h>

#define M 27

#define L 134217728

#define B 16777216

#define F 2000

#define MAXPRIME 17393

/* F=10, MAXPRIME=31

F=100, MAXPRIME=547

F=1000, MAXPRIME=7927

F=2000, MAXPRIME=17393 */

void main(void)

{

unsigned char bitconst[8] =

{0x80, 0x40, 0x20, 0x10, 0x08, 0x04, 0x02, 0x01};

unsigned long j = 0, k;

unsigned short i;

unsigned char *x;

FILE *fp;

unsigned short *prime;

prime = (unsigned short *) malloc((F+1) * sizeof(unsigned

short));

unsigned char *c;

c = (unsigned char *) malloc((MAXPRIME+1) * sizeof(unsigned

char));

for (i = 1; i <= MAXPRIME; i++)

c[i] = 1;

c[1] = 0;

prime[0] = 2;

do

{

for (i = prime[j]; i <= MAXPRIME; i += prime[j])

c[i] = 0;

i = prime[j] + 1;

while (c[i] == 0)

i++;

prime[++j] = i;

}

while (j < F);

free(c);

Page 5

KA

MA

WO

TA

N v

ol.

1,

no. 2

, M

ay

20

07

86

x = (unsigned char *) malloc(B * sizeof(unsigned char));

x[0] = 0x69;

x[1] = ~x[0];

for (i = 1; i < (M-3); i++)

{

k = 1 << i; /* k = 2^i */

for (j = 0; j < k; j++)

x[k+j] = ~x[j];

}

for (i = 1; i <= F; i++)

for (j = prime[i]-1; j < L; j += prime[i])

x[j/8] ^= bitconst[j%8];

fp = fopen("t.dat", "wb");

fwrite(x, B, 1, fp);

fclose(fp);

}

App

endi

x B

/* source code for Microsoft Visual C++ 6.0 */

#include <stdio.h>

#include <stdlib.h>

#define M 27

#define L 134217728

#define B 16777216

#define F 2000

#define MAXPRIME 17393

/* F=10, MAXPRIME=31

F=100, MAXPRIME=547

F=1000, MAXPRIME=7927

F=2000, MAXPRIME=17393

L is the length of the sequence in bits; L = 2^M

B is the size of the sequence in bytes

F is the number of (odd) primes to use

MAXPRIME is the Fth odd prime (the (F+1)th prime) */

void main(void)

{

unsigned char bitconst[8] =

{0x80, 0x40, 0x20, 0x10, 0x08, 0x04, 0x02, 0x01};

unsigned long j = 0, k;

unsigned short i;

unsigned char *x;

FILE *fp;

/* prime[0] = 2; prime[1] = 3; the Fth odd prime is prime[F] */

unsigned short *prime;

prime = (unsigned short *) malloc((F+1) * sizeof(unsigned

short));

unsigned char *c;

Col

lege

of

Art

s a

nd

Sci

en

ces,

Ate

neo

de

Na

ga

Un

ive

rsi

ty

87

c = (unsigned char *) malloc((MAXPRIME+1) * sizeof(unsigned

char));

/* data types might need to be changed if F is increased */

for (i = 1; i <= MAXPRIME; i++)

c[i] = 1;

c[1] = 0;

prime[0] = 2;

do

{

for (i = prime[j]; i <= MAXPRIME; i += prime[j])

c[i] = 0;

i = prime[j] + 1;

while (c[i] == 0)

i++;

prime[++j] = i;

}

while (j < F);

free(c);

/* create Kolakoski word */

x = (unsigned char *) malloc(B * sizeof(unsigned char));

x[0] = 0x34;

for (j = 1; j < B; j++)

x[j] = 0x00;

k = 8;

for (j = 5; k < L; j++)

{

x[k/8] |= bitconst[k%8];

k++;

if ((x[j/8] & bitconst[j%8]) == 0x00)

{

x[k/8] |= bitconst[k%8];

k++;

}

j++;

k++;

if ((x[j/8] & bitconst[j%8]) == 0x00)

k++;

}

/* invert bits that are multiples of the first F odd primes */

for (i = 1; i <= F; i++)

for (j = prime[i]-1; j < L; j += prime[i])

x[j/8] ^= bitconst[j%8];

fp = fopen("k.dat", "wb");

fwrite(x, B, 1, fp);

fclose(fp);

}