Find the next number in this series:

series/factorial.towers New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Find the next number in this series:

0, 1, 2, 720!

The ! is factorial, not punctuation.  It's from a book by Douglas
Hofstadter [Editor: "Hofstader" in the original], _Fluid Concepts and Creative Analogies_.

-- Rachel <rhysel@aol.com>, April 1995

Show solution

The n-th term of the sequence is the number n-1 followed by n-1
factorial signs.  So it starts with 0 followed by no factorials, which
is 0, then goes 1! = 1, 2!! = 2! = 2, 3!!! = 6!! = 720!, and so on.  So
the answer is 4!!!! = 24!!!, which is pretty big!  The problem would be
more difficult to solve, if you did not give the clue of putting a
factorial sign in the problem - i.e. it should go, what is the next
term in the sequence:

0, 1, 2,
260121894356579510020490322708104361119152187501694578572754183785083563115694\
738224067857795813045708261992057589224725953664156516205201587379198458774083\
252910524469038881188412376434119195104550534665861624327194019711390984553672\
727853709934562985558671936977407000370043078375899742067678401696720784628062\
922903210716166986726054898844551425719398549944893959449606404513236214026598\
619307324936977047760606768067017649166940303481996188145562519559256691883082\
551494294759653727484562462882423452659778973774089646655399243592878621251596\
748322097602950569669992728467056374713753301924831358707612541268341586012944\
756601145542074958995256354306828863463108496565068277155299625679084523570255\
218622235813001670083452344323682193579318470195651072978180435417389056072742\
804858399591972902172661229129842051606757903623233769945396419147517556755769\
539223380305682530859997744167578435281591346134039460490126954202883834710136\
373382448450666009334848444071193129253769465735433737572477223018153403264717\
753198453734147867432704845798378661870325740593892421570969599463055752106320\
326349320922073832092335630992326750440170176057202601082928804233560664308988\
871029738079757801305604957634283868305719066220529117482251053669775660302957\
404338798347151855260280533386635713910104633641976909739743228599421983704697\
910995630338960467588986579571117656667003915674815311594398004362539939973120\
306649060132531130471902889849185620376666916446879112524919375442584589500031\
156168297430464114253807489728172337595538066171980140467793561479363526626568\
333950976000000000000000000000000000000000000000000000000000000000000000000000\
000000000000000000000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000
?

-- Dr D F Holt <mareg@csv.warwick.ac.uk>, April 1995

The sequence appears to be 1!!!, 2!!!, 3!!! (= 720!), so the fourth
term would be 4!!!.

-- Stacy Friedman <safriedm@husc7.harvard.edu>

Ouch!  Ow!  Blammo!  Incorrect! :)

0!!! is 1, not 0.

-- Michael Soss <soss@course3.harvard.edu>, April 1995