Let a and c be integers. "Shew that" F := 1, F := a c - 1, F := 0 1 n 2 (2 n + a - 3) ((2 c - 1) (2 n + a - 4) (2 n + a - 2) + (a - 2) ) F n - 1 (----------------------------------------------------------------------- 2 - (n - 1) (n + a - 3) (2 n + a - 2) F )/(n (n + a - 2) (2 n + a - 4)) n - 2 (note cubic denominator) is an integer sequence. Distantly related: it appears that [ n + 4 n + 5 ] [ - n, -----, ----- | ] F [ 2 2 | -4] = 1,6,57,701,10147,164317,2888282,54047434, 3 2 [ | ] [ 3, 2 ] is an integer sequence. The smallest prime factors go {},2,3,701,73,37,2,2,7,2,3,3,5,2,23,37,65306671610636210891,2, 31,3,3,19,2,2,3,2,3,7,3,3,2,2,7,2,67,5,... . Likewise, [ n + 5 n + 6 ] [ - n, -----, ----- | ] 4 F [ 2 2 | -4] = 4,25,228,2620,35164,527663,8613004, 3 2 [ | ] [ 4, 2 ] [ n + 6 n + 7 ] [ - n, -----, ----- | ] 15 F [ 2 2 | -4] = 15,99,891,9825,125085,1772775,27303603, 3 2 [ | ] [ 5, 2 ] [ n + 7 n + 8 ] [ - n, -----, ----- | ] 5 F [ 2 2 | -4] = 5,29,230,2260,25921,334105,4717653, 3 2 [ | ] [ 5, 3 ] More generally, [ n + k + 1 n + k + 2 ] [ n + k + 2 n + k + 3 ] [-n,---------,---------| ] [-n,---------,---------| ] F [ 2 2 |-4] & F [ 2 2 |-4] 3 2[ | ] 3 2[ | ] [ k, 2 ] [ k, 3 ] appear to have bounded denominators. --rwg Does OEIS list sequences not proven integral?
[CCing seqfan]
R. William Gosper [...]
Wow! These are like Michael Somos's unreasonably-integral sequences on steroids.
Does OEIS list sequences not proven integral?
The OEIS contains quite a few sequences that depend on open questions. And, fortunately, being online it is easily revised. Just provide a comment like "Not proven to be integral for all N". This can later be updated when it is proven, or else the definition can be be fudged to something like "numerator of...".
mlb>Wow! These are like Michael Somos's unreasonably-integral sequences
on steroids.
Shallower, I fear. The recurrence I gave for F coincides with a three- term recurrence for a terminating 2F1 with entirely integral terms. And, to my mild embarrassment, the 3F2[-4] = 1,6,57,701,10147,164317,2888282, can be written n ==== \ n (n + 2 i + 3)! > 2 ( ) --------------------------, / i (i + 1)! (i + 2)! (n + 3)! ==== i = 0 which likewise appears to be termwise integral. (It would be obvious (and even) if that numerator were (n+2i+6)!, e.g. We'd have a nice, Pascaloid recurrence.) So why is this summand an integer. Or is it? --rwg
The integerhood of a 3F2 summand I mentioned follows from the conjecture that n /===\ k - j + 1 k + 2 j + 2 k + 2 j + 3 | | (1 - q ) (1 - q ) (1 - q ) | | ------------------------------------------------------ | | j j + 1 j + 2 j = 1 (1 - q ) (1 - q ) (1 - q ) is a polynomial in q for all n,k >0. This seems to be hard!? I'll bet there's a theorem that for an expression like this, you only need to check a dozen or so consecutive n and k --rwg Chorasmian maraschino harmonicas inspissate antisepsis.
----- Original Message ----- From: "R. William Gosper" <rwg@tc.spnet.com> To: <math-fun@mailman.xmission.com> Sent: Thursday, October 09, 2003 10:15 AM Subject: Re: [math-fun] nonobviously integer sequence
Chorasmian maraschino harmonicas inspissate antisepsis.
Good, if a bit farfetched as well as slightly unlikely, truth-wise. Now try to find permutations of an alternate spelling, "Khwarezmian."
Copied Bill's sentence to Andrew Bremner, who replied with Sacrificial-stones reclassifications capillarity piratically altered related mountaineer enumeration. R. On Thu, 9 Oct 2003, Steve Gray wrote:
----- Original Message ----- From: "R. William Gosper" <rwg@tc.spnet.com> To: <math-fun@mailman.xmission.com> Sent: Thursday, October 09, 2003 10:15 AM Subject: Re: [math-fun] nonobviously integer sequence
Chorasmian maraschino harmonicas inspissate antisepsis.
Good, if a bit farfetched as well as slightly unlikely, truth-wise. Now try to find permutations of an alternate spelling, "Khwarezmian."
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participants (4)
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Marc LeBrun -
R. William Gosper -
Richard Guy -
Steve Gray