On Fri, Jun 5, 2020 at 9:15 AM Bill Gosper <billgosper@gmail.com> wrote:
Simon Plouffe's old http://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=1.64163256065515386629&... told me that
In[202]:= Dot @@ Table[{{1/2 + 2^-n, -1/2}, {-1/2, 1/2}}, {n, ∞}]
= {{1,-1},{1-𝜗, 𝜗-1}]
(if only Mathematica supported infinite Tables) where 𝜗 =
In[186]:= Sum[2^-Binomial[n, 2], {n, ∞]}]
Out[186]= EllipticTheta[2, 0, 1/√2]/2^(7/8)
I.e., this product converges to something interesting even though the determinant approaches zero.
I can't wait to try other things, but agonizingly, Simon's website seems up only a few seconds every few minutes! —rwg
This problem turned out to be bagsucking Safari falsely insisting the wayback site was down! A day or two ago, Safari Gmail utterly and quietly reamed me by replacing a very hardwon graphic with a load of excrement about enabling JavaScript! Tim Cook is not Steve Jobs. Anyway, this all left only time to find Dot[Table[{{1/2+2^-n,-(1/2)},{-(1/2),1/2}},{n,2,\[Infinity]}]]=={ {4-2^(1/8) EllipticTheta[2,0,1/Sqrt[2]],-4+2^(1/8) EllipticTheta[2,0,1/Sqrt[2]]}, {6-2 2^(1/8) EllipticTheta[2,0,1/Sqrt[2]],-6+2 2^(1/8) EllipticTheta[2,0,1/Sqrt[2]]}} Plouffe finds these halvish ones as binary generating constants for OEISequences. It even found A007401: 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22 " Add n-1 to n-th term of 'n appears n times' sequence (A002024 <http://oeis.org/A002024>). " which seemed very strange until I realized it was the complement of a quadratic sequence. So if Plouffe fails or actually does go down, we can directly appeal to OEIS with Position[RealDigits[...,2],1] —rwg