There seems to be just one more of these, namely when x^3+x=-1, e.g., x == -(2/(3 (-9 + Sqrt[93])))^(1/3) + (1/2 (-9 + Sqrt[93]))^(1/3)/3^( 2/3) == -((2 Sinh[1/3 ArcSinh[(3 Sqrt[3])/2]])/Sqrt[3]) (for which Mma won't say True except numerically), the Somos4 sequence 1,1,-1,x,-x^3,-x^5,0,x^9,,x^11,-x^13,x^16,-x^20,-x^24,0,x^32,x^36,-x^40,x^45,-x^51,-x^57,0,... is a period 7 sequence times a supergeometric in x. The ?s in this progression of exponents 0,0,0,1,3,5,?,9,11,13,16,20,24,?,32,... pose an interesting indexing|lookup problem for OEIS. --rwg ISC needs a few zillion RootOfs. On Sat, May 29, 2010 at 9:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Thu, May 27, 2010 at 9:00 PM, Bill Gosper <billgosper@gmail.com> wrote:
Let's call sequences growing like x^n^2 "supergeometric". The Somos recurrence s[n]:=(s[n-1]*s[n-3]+s[n-2]^2)/s[n-4], for s[1],... = 1,1,1,1,..., grows supergeometrically. But if we initialize with 1,1,-1,-2/3,..., we get a supergeometrically *shrinking* sequence = a period 8 sequence times 3^-(n^2/16):
s[n] = -2*%i^(n^2+1)*q^(n^2/64)*theta[1](%i*n*log(q)/8+n*%pi/2,q)/ theta[4](0,q)*3^(n^2/16)
2 2 n + 1 n /64 i n log(q) n pi 2 i q theta (---------- + ----, q) 1 8 2 s = - ------------------------------------------ n 2 n /16 theta (0, q) 3 4
= 1,1,-1,-2/3,1/3,1/9,-1/27,0,1/243,-1/729,-1/2187,2/3^9,...
where the simplest equation I've found for q is
3^(1/4)*theta[4](0,q) = -theta[2](0,%i*q^(1/4))*%e^(7*i*%pi/8)
7 i pi ------ 1/4 1/4 8 3 theta (0, q) = - theta (0, i q ) e 4 2
Can we do better?
Similarly, the period 8 sequence (1,1,-1,0,1,-1,-1,0)* comes from
s[n] = theta[1](n*%pi/4,q)/(theta[2](0,q)*%i^(n^2/4-1))
n pi theta (----, q) 1 4 s = -------------------- n 2 n -- - 1 4 theta (0, q) i 2
with q a root of
theta[2](0,q) = theta[2](0,%i*q)*sqrt(%i)/sqrt(2)
theta (0, i q) sqrt(i) 2 theta (0, q) = ---------------------- 2 sqrt(2)
And then there's the even simpler period 5 sequence (1,1,-1,-1,0)* from s[n] = theta[1](2*n*%pi/5,q)/theta[1](%pi/5,q)
2 n pi theta (------, q) 1 5 s = ----------------- n pi theta (--, q) 1 5
with q defined by
theta[1](%pi/5,q) = theta[1](2*n*%pi/5,q)
pi 2 n pi theta (--, q) = theta (------, q). 1 5 1 5
This theta is presumably algebraic? That would bode ill for closedform q. The only thetas I know how to invert are algebraic times gamma(s).
Of course, all of these can be expressed as a pile of trigs But not when we generalize to 1,1,-1,x,... . Can anyone say why this makes polynomials in x?
With x = phi and -1/phi we get period 6 sequences times supergeometrics. Dividing their difference by rt5 gives 6 interlaced fibonacci(qudratic)s: http://gosper.org/fibsom.pdf . There must be a nice relation between the p and q nomes.
--rwg