13 October 2007

What comes next: 1, 2, 5, ...

What comes next in this sequence: 1, 2, 5, ...?

I know of at least seven reasonable answers: 10, 11, 12, 13, 14, or 15.

Here are explanations for each of them. Throughout, I'll denote the nth term of each sequence by an, where indexing starts at 1. (So a3 = 5, and we're looking for a4. This indexing convention is awkward for the first couple cases but agrees with the conventional indexing for Catalan and Bell numbers.)

1, 2, 5, 10, ... -- an = (n-1)2 + 1.

1, 2, 5, 11, ... -- $a_n = {n-1 \choose 3}$

1, 2, 5, 12, ... -- an = 2an-1 + an-2, where a1 = 1, a2 = 2. This sequence occurs in the convergents of the continued fraction for √2. The first few convergents are

1/1, 3/2, 7/5, 17/12, 41/29

and in general the nth convergent, bn/an, has an = an-1 + bn-1, bn = an + bn-1, with a1 = b1 = 1. An explicit formula for these is given by

a_n = {1 \over 2\sqrt{2}} \left[ \left( 1+\sqrt{2} \right)^n + \left(1- \sqrt{2} \right)^n \right]


1, 2, 5, 13, ... -- alternate members of the Fibonacci sequence, which begins 1, 1, 2, 3, 5, 8, 13. If you don't like referring to another sequence, this one can be defined by the recurrence an = 3an-1 - an-2, where a1 = 1, a2 = 2. An exact formula for the nth Fibonacci number is fn = (φn - (-&phi)-n)/√5, where φ = (-1+√5)/2; then an = f2n-1.

1, 2, 5, 14, ... -- the Catalan numbers. These satisfy $a_n = {1 \over 2n+1} {2n \choose n}$. The Catalan numbers count basically everything. Okay, not everything. But in Richard Stanley's book Enumerative Combinatorics, he gives 66 sets counted by Catalan numbers; he has an ever-expanding list of them on his web site, which is currently up to 162, numbered (a), (b), (c), ..., (z), (aa), ..., (zz), (aaa), ... (zzz), (a4), ..., (z4), ..., (f7). This list may hold the prize for "longest list of things indicated by letters". The canonical example of "things counted by Catalan numbers" is binary trees, counted by number of leaves, and most of these other things can eventually be reduced to that case. The Catalan numbers satisfy the nice recurrence

a_{n+1} = \sum_{k=0}^n a_k a_{n-k}


which explains why they're so prevalent; they correspond to combinatorial objects that can be generated recursively, where each object of size n+1 consists of a pair of similar objects whose sizes together add up to n.

1, 2, 5, 15, ... -- the Bell numbers. These are the number of set partitions of the set {1, 2, ..., n}; that is, the number of ways to arrange n balls in some boxes. For four balls, we have the partitions

1234, 123/4, 124/3, 134/2, 234/1, 12/34, 13/24, 14/23, 12/3/4, 13/2/4, 14/2/3, 23/1/4, 24/1/3, 34/1/2, 1/2/3/4

where slashes separate the different boxes. There's no explicit formula for these, but they satisfy the recursion

a_{n+1} = \sum_{k=0}^n {n \choose k} a_k


which can be justified as follows. If we want to count set partitions of {1, 2, ..., n+1}, we can count them by the number of elements in the block containing n+1. The number of set partitions of {1, 2, ..., n+1} where n+1 is in a block of size j+1 is ${n \choose j} a_{n-j}$; we can pick the j elements which are in the same block as n+1 and then make a set partition out of the remaining n-j elements. Thus

a_n = \sum_{j=0}^n {n \choose j} a_{n-j}


and letting k = n-j and recalling ${n \choose n-k} = {n \choose k}$ gives the desired result.

These sequences diverge from each other quickly; the next term of each sequence is 17, 21, 29, 34, 42, and 52 respectively. And the growth rates of the six sequences in question are quite different; roughly, the sequences given here grow like n2, n3/6, (1+&sqrt;(2))n, ((3+&sqrt;(5))/2)n, 4n/(π n3)1/2, and (really fast). Their generating functions also look quite different. For the first five sequences, we have ordinary generating functions which converge in some neighborhood of the origin, which are

{z(z-2z^2-1) \over (z-1)^3}, {z(1-2z+3z^2-z^3) \over (1-z)^4}, {z \over 1-z-z^2}, {z-z^2 \over 1-3z+z^2}, {1-2z-\sqrt{1-4z} \over 2z}


but the last one doesn't even have a convergent ordinary generating function; its exponential generating function is exp(exp(z)-1). There's a subtle difference among the ordinary generating functions as well -- the first two have singularities at z=1, so the sequences they represent grow only as fast as powers of n, while the last three have singularities closer to the origin than z=1, so the sequences they represent grow roughly exponentially.

This is all an example of what I've heard called "the law of small numbers" -- the fact that there just aren't enough small numbers to avoid numerical coincidences like these.

37 comments:

D. Eppstein said...

BTW, 1, 2, 5, 12, ... are the Pell numbers, though they were known long before Euler incorrectly credited Pell with them.

John Armstrong said...

I don't think this has anything to do with the "law of small numbers". If you give me any number of entries from a sequence, I can rationalize any next number in some way or another.

It took me a long time to realize this, but the important thing about combinatorics actually isn't the sequence. It's when two different structures give rise to the same sequence.

As you said, the Catalan numbers count pretty much everything. For two examples, they count Temperley-Lieb diagrams and they count the dimensions of irreducible $SL_q(2)$ modules. From here, we get the idea to look for Temperley-Lieb recoupling theory, and a whole field of mathematics is born.

.mau. said...

1, 2, 5 is much too short a sequence to say anything about its next terms.
What about 26? it follows from f_{n+1} = (f_n)^2 + 1. Or 3126? it follows from f_{n+1} = (f_n)^(f_n) + 1 :-)

Michael Cassidy said...

So I don't see how any n<4 works in: C(3, n-1); your second example.

Qiaochu Yuan said...

I wouldn't call the Fibonacci-Catalan-Bell triplet coincidental: there's a "reason" the first three terms of each are the same.

The alternating Fibonacci numbers count closed walks from one end of the path graph P_4 to itself, and the Catalan numbers count the number of closed walks from one end of the infinite path graph P_{\infty} to itself (this is the Dyck word interpretation). So the Fibonacci numbers "approximate" the Catalan numbers. You can get a sequence of such approximations by taking P_n for larger n (since the first n-1 terms will always agree), and I believe these are the same as the approximations you get by truncating the continued fraction for the power series of the Catalan numbers.

Similarly, just as the Bell numbers count partitions, the Catalan numbers count non-crossing partitions. For small values of n, all partitions are non-crossing, hence the agreement.

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[url= http://www.projectopus.com/user/58289 ] KATE MARA NAKED [/url]
OLSEN TWINS VIRGINS
[url=http://www.projectopus.com/user/58293]OLSEN TWINS VIRGINS[/url]
OLSEN TWINS VIRGINS
[url= http://www.projectopus.com/user/58293 ] OLSEN TWINS VIRGINS [/url]

Anonymous said...

ASHLEY HARKLEROAD NUDE
[url=http://www.projectopus.com/user/58295]ASHLEY HARKLEROAD NUDE[/url]
ASHLEY HARKLEROAD NUDE
[url= http://www.projectopus.com/user/58295 ] ASHLEY HARKLEROAD NUDE [/url]
NIURKA DESNUDA
[url=http://www.projectopus.com/user/58297]NIURKA DESNUDA[/url]
NIURKA DESNUDA
[url= http://www.projectopus.com/user/58297 ] NIURKA DESNUDA [/url]
PINAY SEX SCANDAL DOWNLOAD
[url=http://www.projectopus.com/user/58299]PINAY SEX SCANDAL DOWNLOAD[/url]
PINAY SEX SCANDAL DOWNLOAD
[url= http://www.projectopus.com/user/58299 ] PINAY SEX SCANDAL DOWNLOAD [/url]
DISNEY'S RAVEN SYMONE NUDE
[url=http://www.projectopus.com/user/58303]DISNEY'S RAVEN SYMONE NUDE[/url]
DISNEY'S RAVEN SYMONE NUDE
[url= http://www.projectopus.com/user/58303 ] DISNEY'S RAVEN SYMONE NUDE [/url]
ASHLEY HARKLEROAD PLAYBOY
[url=http://www.projectopus.com/user/58305]ASHLEY HARKLEROAD PLAYBOY[/url]
ASHLEY HARKLEROAD PLAYBOY
[url= http://www.projectopus.com/user/58305 ] ASHLEY HARKLEROAD PLAYBOY [/url]

Anonymous said...

ELLEN DEGENERES GAY
[url=http://videoexclf1.vidiLife.com]ELLEN DEGENERES GAY[/url]
ELLEN DEGENERES GAY
[url= http://videoexclf1.vidiLife.com ] ELLEN DEGENERES GAY [/url]
SCARLETT JOHANSSON BUTT
[url=http://videoexclf7.vidiLife.com]SCARLETT JOHANSSON BUTT[/url]
SCARLETT JOHANSSON BUTT
[url= http://videoexclf7.vidiLife.com ] SCARLETT JOHANSSON BUTT [/url]
PAIGE DAVIS STRIPPING
[url=http://videoexclf8.vidiLife.com]PAIGE DAVIS STRIPPING[/url]
PAIGE DAVIS STRIPPING
[url= http://videoexclf8.vidiLife.com ] PAIGE DAVIS STRIPPING [/url]
STACY KEIBLER IN A BIKINI
[url=http://videoexclf9.vidiLife.com]STACY KEIBLER IN A BIKINI[/url]
STACY KEIBLER IN A BIKINI
[url= http://videoexclf9.vidiLife.com ] STACY KEIBLER IN A BIKINI [/url]
BREASTS CELEBRITIES
[url=http://videoexclf10.vidiLife.com]BREASTS CELEBRITIES[/url]
BREASTS CELEBRITIES
[url= http://videoexclf10.vidiLife.com ] BREASTS CELEBRITIES [/url]

Anonymous said...

JOVENES DESNUDAS
[url=http://videoexclf16.vidiLife.com]JOVENES DESNUDAS[/url]
JOVENES DESNUDAS
[url= http://videoexclf16.vidiLife.com ] JOVENES DESNUDAS [/url]
KENDRA WILKINSON UNCENSORED
[url=http://videoexclf17.vidiLife.com]KENDRA WILKINSON UNCENSORED[/url]
KENDRA WILKINSON UNCENSORED
[url= http://videoexclf17.vidiLife.com ] KENDRA WILKINSON UNCENSORED [/url]
NIURKA MARCOS DESNUDA
[url=http://videoexclf18.vidiLife.com]NIURKA MARCOS DESNUDA[/url]
NIURKA MARCOS DESNUDA
[url= http://videoexclf18.vidiLife.com ] NIURKA MARCOS DESNUDA [/url]
GALILEA MONTIJO DESNUDA
[url=http://videoexclf19.vidiLife.com]GALILEA MONTIJO DESNUDA[/url]
GALILEA MONTIJO DESNUDA
[url= http://videoexclf19.vidiLife.com ] GALILEA MONTIJO DESNUDA [/url]
DESNUDAS EN LA PLAYA
[url=http://videoexclf20.vidiLife.com]DESNUDAS EN LA PLAYA[/url]
DESNUDAS EN LA PLAYA
[url= http://videoexclf20.vidiLife.com ] DESNUDAS EN LA PLAYA [/url]

Anonymous said...

MAYRA VERONICA NUDE
[url=http://my.curse.com/members/KRYSTLE5.aspx]MAYRA VERONICA NUDE[/url]
MAYRA VERONICA NUDE
[url= http://my.curse.com/members/KRYSTLE5.aspx ] MAYRA VERONICA NUDE [/url]
COLOMBIANAS DESNUDAS
[url=http://my.curse.com/members/KYLA4.aspx]COLOMBIANAS DESNUDAS[/url]
COLOMBIANAS DESNUDAS
[url= http://my.curse.com/members/KYLA4.aspx ] COLOMBIANAS DESNUDAS [/url]
HANNAH MONTANA NAKED
[url=http://my.curse.com/members/KYLEE3.aspx]HANNAH MONTANA NAKED[/url]
HANNAH MONTANA NAKED
[url= http://my.curse.com/members/KYLEE3.aspx ] HANNAH MONTANA NAKED [/url]
TARA REID BODY SHOTS
[url=http://my.curse.com/members/KYMBERLY3.aspx]TARA REID BODY SHOTS[/url]
TARA REID BODY SHOTS
[url= http://my.curse.com/members/KYMBERLY3.aspx ] TARA REID BODY SHOTS [/url]
CELEBRITIES EXPOSED THONGS
[url=http://my.curse.com/members/KYUNG3.aspx]CELEBRITIES EXPOSED THONGS[/url]
CELEBRITIES EXPOSED THONGS
[url= http://my.curse.com/members/KYUNG3.aspx ] CELEBRITIES EXPOSED THONGS [/url]

Anonymous said...

FAMOSAS MEXICANAS DESNUDAS
[url=http://www.projectopus.com/user/58529]FAMOSAS MEXICANAS DESNUDAS[/url]
FAMOSAS MEXICANAS DESNUDAS
[url= http://www.projectopus.com/user/58529 ] FAMOSAS MEXICANAS DESNUDAS [/url]
LINDSAY LOHAN CLEAVAGE
[url=http://www.projectopus.com/user/58531]LINDSAY LOHAN CLEAVAGE[/url]
LINDSAY LOHAN CLEAVAGE
[url= http://www.projectopus.com/user/58531 ] LINDSAY LOHAN CLEAVAGE [/url]
NENAS DESNUDAS
[url=http://www.projectopus.com/user/58533]NENAS DESNUDAS[/url]
NENAS DESNUDAS
[url= http://www.projectopus.com/user/58533 ] NENAS DESNUDAS [/url]
GLORIA TREVI DESNUDA
[url=http://www.projectopus.com/user/58535]GLORIA TREVI DESNUDA[/url]
GLORIA TREVI DESNUDA
[url= http://www.projectopus.com/user/58535 ] GLORIA TREVI DESNUDA [/url]
SOFIA VERGARA DESNUDA
[url=http://www.projectopus.com/user/58539]SOFIA VERGARA DESNUDA[/url]
SOFIA VERGARA DESNUDA
[url= http://www.projectopus.com/user/58539 ] SOFIA VERGARA DESNUDA [/url]

Anonymous said...

OLSEN TWINS CAMEL TOE
[url=http://www.projectopus.com/user/58541]OLSEN TWINS CAMEL TOE[/url]
OLSEN TWINS CAMEL TOE
[url= http://www.projectopus.com/user/58541 ] OLSEN TWINS CAMEL TOE [/url]
NEXT DOOR NIKKI VIDEO
[url=http://www.projectopus.com/user/58543]NEXT DOOR NIKKI VIDEO[/url]
NEXT DOOR NIKKI VIDEO
[url= http://www.projectopus.com/user/58543 ] NEXT DOOR NIKKI VIDEO [/url]
VIDA GUERRA BUTT
[url=http://www.projectopus.com/user/58545]VIDA GUERRA BUTT[/url]
VIDA GUERRA BUTT
[url= http://www.projectopus.com/user/58545 ] VIDA GUERRA BUTT [/url]
KENDRA WILKINSON GALLERY
[url=http://www.projectopus.com/user/58547]KENDRA WILKINSON GALLERY[/url]
KENDRA WILKINSON GALLERY
[url= http://www.projectopus.com/user/58547 ] KENDRA WILKINSON GALLERY [/url]
LIL KIM UNCENSORED
[url=http://www.projectopus.com/user/58549]LIL KIM UNCENSORED[/url]
LIL KIM UNCENSORED
[url= http://www.projectopus.com/user/58549 ] LIL KIM UNCENSORED [/url]

Anonymous said...

GORDAS DESNUDAS
[url=http://www.projectopus.com/user/58569]GORDAS DESNUDAS[/url]
GORDAS DESNUDAS
[url= http://www.projectopus.com/user/58569 ] GORDAS DESNUDAS [/url]
SCARLETT JOHANSSON GALLERY
[url=http://www.projectopus.com/user/58571]SCARLETT JOHANSSON GALLERY[/url]
SCARLETT JOHANSSON GALLERY
[url= http://www.projectopus.com/user/58571 ] SCARLETT JOHANSSON GALLERY [/url]
SARAH PALIN NAKED
[url=http://www.projectopus.com/user/58573]SARAH PALIN NAKED[/url]
SARAH PALIN NAKED
[url= http://www.projectopus.com/user/58573 ] SARAH PALIN NAKED [/url]
MILA KUNIS NAKED
[url=http://www.projectopus.com/user/58575]MILA KUNIS NAKED[/url]
MILA KUNIS NAKED
[url= http://www.projectopus.com/user/58575 ] MILA KUNIS NAKED [/url]
LAURA SAN GIACOMO NUDE
[url=http://www.projectopus.com/user/58577]LAURA SAN GIACOMO NUDE[/url]
LAURA SAN GIACOMO NUDE
[url= http://www.projectopus.com/user/58577 ] LAURA SAN GIACOMO NUDE [/url]

Anonymous said...

LEIGHTON MEESTER NUDE
[url=http://www.projectopus.com/user/58591]LEIGHTON MEESTER NUDE[/url]
LEIGHTON MEESTER NUDE
[url= http://www.projectopus.com/user/58591 ] LEIGHTON MEESTER NUDE [/url]
KATE BOSWORTH NUDE
[url=http://www.projectopus.com/user/58593]KATE BOSWORTH NUDE[/url]
KATE BOSWORTH NUDE
[url= http://www.projectopus.com/user/58593 ] KATE BOSWORTH NUDE [/url]
AISHWARYA RAI BIKINI
[url=http://www.projectopus.com/user/58595]AISHWARYA RAI BIKINI[/url]
AISHWARYA RAI BIKINI
[url= http://www.projectopus.com/user/58595 ] AISHWARYA RAI BIKINI [/url]
RAVEN SYMONE BREAST
[url=http://www.projectopus.com/user/58597]RAVEN SYMONE BREAST[/url]
RAVEN SYMONE BREAST
[url= http://www.projectopus.com/user/58597 ] RAVEN SYMONE BREAST [/url]
ELISHA CUTHBERT NAKED
[url=http://www.projectopus.com/user/58599]ELISHA CUTHBERT NAKED[/url]
ELISHA CUTHBERT NAKED
[url= http://www.projectopus.com/user/58599 ] ELISHA CUTHBERT NAKED [/url]

Anonymous said...

SARAH PALIN PORN
[url=http://my.curse.com/members/LAVON4.aspx]SARAH PALIN PORN[/url]
SARAH PALIN PORN
[url= http://my.curse.com/members/LAVON4.aspx ] SARAH PALIN PORN [/url]
JENNIFER MORRISON NUDE
[url=http://my.curse.com/members/LAVONIA2.aspx]JENNIFER MORRISON NUDE[/url]
JENNIFER MORRISON NUDE
[url= http://my.curse.com/members/LAVONIA2.aspx ] JENNIFER MORRISON NUDE [/url]
JENNA JAMESON FUCKING
[url=http://my.curse.com/members/LAVONNE7.aspx]JENNA JAMESON FUCKING[/url]
JENNA JAMESON FUCKING
[url= http://my.curse.com/members/LAVONNE7.aspx ] JENNA JAMESON FUCKING [/url]
ADRIENNE BAILON NUDE PICS
[url=http://my.curse.com/members/LAWANNA.aspx]ADRIENNE BAILON NUDE PICS[/url]
ADRIENNE BAILON NUDE PICS
[url= http://my.curse.com/members/LAWANNA.aspx ] ADRIENNE BAILON NUDE PICS [/url]
MUJERES DESNUDA
[url=http://my.curse.com/members/LAWRENCE4.aspx]MUJERES DESNUDA[/url]
MUJERES DESNUDA
[url= http://my.curse.com/members/LAWRENCE4.aspx ] MUJERES DESNUDA [/url]

Anonymous said...

EMMA WATSON PUSSY
[url=http://my.curse.com/members/LEANDRA3.aspx]EMMA WATSON PUSSY[/url]
EMMA WATSON PUSSY
[url= http://my.curse.com/members/LEANDRA3.aspx ] EMMA WATSON PUSSY [/url]
CELEBRITY SEE THRU
[url=http://my.curse.com/members/LEATRICE3.aspx]CELEBRITY SEE THRU[/url]
CELEBRITY SEE THRU
[url= http://my.curse.com/members/LEATRICE3.aspx ] CELEBRITY SEE THRU [/url]
MAGGIE GYLLENHAAL NUDE
[url=http://my.curse.com/members/LEEANN5.aspx]MAGGIE GYLLENHAAL NUDE[/url]
MAGGIE GYLLENHAAL NUDE
[url= http://my.curse.com/members/LEEANN5.aspx ] MAGGIE GYLLENHAAL NUDE [/url]
JUSTIN TIMBERLAKE NAKED
[url=http://my.curse.com/members/LEIGH6.aspx]JUSTIN TIMBERLAKE NAKED[/url]
JUSTIN TIMBERLAKE NAKED
[url= http://my.curse.com/members/LEIGH6.aspx ] JUSTIN TIMBERLAKE NAKED [/url]
PARIS HILTON JPG
[url=http://my.curse.com/members/LEIGHANN4.aspx]PARIS HILTON JPG [/url]
PARIS HILTON JPG
[url= http://my.curse.com/members/LEIGHANN4.aspx ] PARIS HILTON JPG [/url]

Anonymous said...

EVA MENDEZ NUDE
[url=http://videoexclg6.vidiLife.com]EVA MENDEZ NUDE[/url]
EVA MENDEZ NUDE
[url= http://videoexclg6.vidiLife.com ] EVA MENDEZ NUDE [/url]
KATIE HOLMES NAKED
[url=http://videoexclg7.vidiLife.com]KATIE HOLMES NAKED[/url]
KATIE HOLMES NAKED
[url= http://videoexclg7.vidiLife.com ] KATIE HOLMES NAKED [/url]
MAGGIE Q NUDE
[url=http://videoexclg8.vidiLife.com]MAGGIE Q NUDE[/url]
MAGGIE Q NUDE
[url= http://videoexclg8.vidiLife.com ] MAGGIE Q NUDE [/url]
ERICA DURANCE NUDE
[url=http://videoexclg9.vidiLife.com]ERICA DURANCE NUDE[/url]
ERICA DURANCE NUDE
[url= http://videoexclg9.vidiLife.com ] ERICA DURANCE NUDE [/url]
JESSICA ALBA BUTT
[url=http://videoexclg10.vidiLife.com]JESSICA ALBA BUTT[/url]
JESSICA ALBA BUTT
[url= http://videoexclg10.vidiLife.com ] JESSICA ALBA BUTT [/url]

Anonymous said...

GILLIAN ANDERSON NUDE
[url=http://videoexclg11.vidiLife.com]GILLIAN ANDERSON NUDE[/url]
GILLIAN ANDERSON NUDE
[url= http://videoexclg11.vidiLife.com ] GILLIAN ANDERSON NUDE [/url]
ALI LARTER NUDE
[url=http://videoexclg12.vidiLife.com]ALI LARTER NUDE[/url]
ALI LARTER NUDE
[url= http://videoexclg12.vidiLife.com ] ALI LARTER NUDE [/url]
BRITNEY SPEARS DESNUDA
[url=http://videoexclg13.vidiLife.com]BRITNEY SPEARS DESNUDA[/url]
BRITNEY SPEARS DESNUDA
[url= http://videoexclg13.vidiLife.com ] BRITNEY SPEARS DESNUDA [/url]
FILIPINA CELEBRITIES
[url=http://videoexclg14.vidiLife.com]FILIPINA CELEBRITIES[/url]
FILIPINA CELEBRITIES
[url= http://videoexclg14.vidiLife.com ] FILIPINA CELEBRITIES [/url]
FREE CELEBRITY SEX VIDEOS
[url=http://videoexclg15.vidiLife.com]FREE CELEBRITY SEX VIDEOS[/url]
FREE CELEBRITY SEX VIDEOS
[url= http://videoexclg15.vidiLife.com ] FREE CELEBRITY SEX VIDEOS [/url]

Anonymous said...

CELEBRITY NIP SLIPS
[url=http://my.curse.com/members/LESHA7.aspx]CELEBRITY NIP SLIPS[/url]
CELEBRITY NIP SLIPS
[url= http://my.curse.com/members/LESHA7.aspx ] CELEBRITY NIP SLIPS [/url]
SELMA BLAIR NUDE
[url=http://my.curse.com/members/LESLEE3.aspx]SELMA BLAIR NUDE[/url]
SELMA BLAIR NUDE
[url= http://my.curse.com/members/LESLEE3.aspx ] SELMA BLAIR NUDE [/url]
BILLIE PIPER NUDE
[url=http://videoexclg1.vidiLife.com]BILLIE PIPER NUDE[/url]
BILLIE PIPER NUDE
[url= http://videoexclg1.vidiLife.com ] BILLIE PIPER NUDE [/url]
GALLERY OF PARIS HILTON
[url=http://videoexclg4.vidiLife.com]GALLERY OF PARIS HILTON[/url]
GALLERY OF PARIS HILTON
[url= http://videoexclg4.vidiLife.com ] GALLERY OF PARIS HILTON [/url]
MILEY CYRUS BOOBS
[url=http://videoexclg5.vidiLife.com]MILEY CYRUS BOOBS[/url]
MILEY CYRUS BOOBS
[url= http://videoexclg5.vidiLife.com ] MILEY CYRUS BOOBS [/url]

Anonymous said...

NATALIE PORTMAN NAKED
[url=http://videoexclg16.vidiLife.com]NATALIE PORTMAN NAKED[/url]
NATALIE PORTMAN NAKED
[url= http://videoexclg16.vidiLife.com ] NATALIE PORTMAN NAKED [/url]
AMANDA BEARD PLAYBOY
[url=http://videoexclg17.vidiLife.com]AMANDA BEARD PLAYBOY[/url]
AMANDA BEARD PLAYBOY
[url= http://videoexclg17.vidiLife.com ] AMANDA BEARD PLAYBOY [/url]
ADRIENNE BAILON NAKED
[url=http://videoexclg18.vidiLife.com]ADRIENNE BAILON NAKED[/url]
ADRIENNE BAILON NAKED
[url= http://videoexclg18.vidiLife.com ] ADRIENNE BAILON NAKED [/url]
NICOLE SCHERZINGER NUDE
[url=http://videoexclg19.vidiLife.com]NICOLE SCHERZINGER NUDE[/url]
NICOLE SCHERZINGER NUDE
[url= http://videoexclg19.vidiLife.com ] NICOLE SCHERZINGER NUDE [/url]
SARAH PALIN NUDE
[url=http://videoexclg20.vidiLife.com]SARAH PALIN NUDE[/url]
SARAH PALIN NUDE
[url= http://videoexclg20.vidiLife.com ] SARAH PALIN NUDE [/url]

Anonymous said...

ALYSON HANNIGAN NUDE
[url=http://videoexclg31.vidiLife.com]ALYSON HANNIGAN NUDE[/url]
ALYSON HANNIGAN NUDE
[url= http://videoexclg31.vidiLife.com ] ALYSON HANNIGAN NUDE [/url]
RACHEL WEISZ NUDE
[url=http://videoexclg32.vidiLife.com]RACHEL WEISZ NUDE[/url]
RACHEL WEISZ NUDE
[url= http://videoexclg32.vidiLife.com ] RACHEL WEISZ NUDE [/url]
LINDSAY LOHAN BREASTS
[url=http://videoexclg33.vidiLife.com]LINDSAY LOHAN BREASTS[/url]
LINDSAY LOHAN BREASTS
[url= http://videoexclg33.vidiLife.com ] LINDSAY LOHAN BREASTS [/url]
JESSICA SIMPSON NIP SLIP
[url=http://videoexclg34.vidiLife.com]JESSICA SIMPSON NIP SLIP[/url]
JESSICA SIMPSON NIP SLIP
[url= http://videoexclg34.vidiLife.com ] JESSICA SIMPSON NIP SLIP [/url]
MILLA JOVOVICH NUDE
[url=http://videoexclg35.vidiLife.com]MILLA JOVOVICH NUDE[/url]
MILLA JOVOVICH NUDE
[url= http://videoexclg35.vidiLife.com ] MILLA JOVOVICH NUDE [/url]

Anonymous said...

ROBBS CELEBRITY OOPS
[url=http://videoexclg36.vidiLife.com]ROBBS CELEBRITY OOPS[/url]
ROBBS CELEBRITY OOPS
[url= http://videoexclg36.vidiLife.com ] ROBBS CELEBRITY OOPS [/url]
CAMERON DIAZ NAKED
[url=http://videoexclg37.vidiLife.com]CAMERON DIAZ NAKED[/url]
CAMERON DIAZ NAKED
[url= http://videoexclg37.vidiLife.com ] CAMERON DIAZ NAKED [/url]
OLSEN TWINS NAKED
[url=http://videoexclg38.vidiLife.com]OLSEN TWINS NAKED[/url]
OLSEN TWINS NAKED
[url= http://videoexclg38.vidiLife.com ] OLSEN TWINS NAKED [/url]
CELEBRITY NUDITY DATABASE
[url=http://videoexclg39.vidiLife.com]CELEBRITY NUDITY DATABASE[/url]
CELEBRITY NUDITY DATABASE
[url= http://videoexclg39.vidiLife.com ] CELEBRITY NUDITY DATABASE [/url]
CELEBRITIES NUDITY
[url=http://videoexclg40.vidiLife.com]CELEBRITIES NUDITY[/url]
CELEBRITIES NUDITY
[url= http://videoexclg40.vidiLife.com ] CELEBRITIES NUDITY [/url]

Anonymous said...

MEG RYAN NUDE
[url=http://videoexclg41.vidiLife.com]MEG RYAN NUDE[/url]
MEG RYAN NUDE
[url= http://videoexclg41.vidiLife.com ] MEG RYAN NUDE [/url]
RAVEN RILEY SEX
[url=http://videoexclg42.vidiLife.com]RAVEN RILEY SEX[/url]
RAVEN RILEY SEX
[url= http://videoexclg42.vidiLife.com ] RAVEN RILEY SEX [/url]
PARIS HILTON BLOW JOB
[url=http://videoexclg43.vidiLife.com]PARIS HILTON BLOW JOB[/url]
PARIS HILTON BLOW JOB
[url= http://videoexclg43.vidiLife.com ] PARIS HILTON BLOW JOB [/url]
ANNA FARIS NUDE
[url=http://videoexclg44.vidiLife.com]ANNA FARIS NUDE[/url]
ANNA FARIS NUDE
[url= http://videoexclg44.vidiLife.com ] ANNA FARIS NUDE [/url]
MARISA TOMEI NUDE
[url=http://videoexclg45.vidiLife.com]MARISA TOMEI NUDE[/url]
MARISA TOMEI NUDE
[url= http://videoexclg45.vidiLife.com ] MARISA TOMEI NUDE [/url]

Anonymous said...

BRAD PITT NUDE
[url=http://videoexclg46.vidiLife.com]BRAD PITT NUDE[/url]
BRAD PITT NUDE
[url= http://videoexclg46.vidiLife.com ] BRAD PITT NUDE [/url]
LUCY PINDER NUDE
[url=http://videoexclg47.vidiLife.com]LUCY PINDER NUDE[/url]
LUCY PINDER NUDE
[url= http://videoexclg47.vidiLife.com ] LUCY PINDER NUDE [/url]
LAURA PREPON NUDE
[url=http://videoexclg48.vidiLife.com]LAURA PREPON NUDE[/url]
LAURA PREPON NUDE
[url= http://videoexclg48.vidiLife.com ] LAURA PREPON NUDE [/url]
LIV TYLER NUDE
[url=http://videoexclg49.vidiLife.com]LIV TYLER NUDE[/url]
LIV TYLER NUDE
[url= http://videoexclg49.vidiLife.com ] LIV TYLER NUDE [/url]
EVA LONGORIA NAKED
[url=http://videoexclg50.vidiLife.com]EVA LONGORIA NAKED[/url]
EVA LONGORIA NAKED
[url= http://videoexclg50.vidiLife.com ] EVA LONGORIA NAKED [/url]