login
A112259
Expansion of x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)).
4
1, 5, 9, 605, 961, 16245, 284089, 29645, 15046641, 101025125, 73222249, 9908816445, 23755748641, 204034140245, 5031349566489, 1965713970605, 219320727489361, 1965265930868805, 374345220088009, 158335559155140125
OFFSET
1,2
COMMENTS
Previous name was: Let p = the golden mean = (1+sqrt(5))/2. A point in 3-space is identified by three numbers t = (a,b,c). f(t) is the product a*b*c. Let t = (-1/p,1,p): using the rules of 'triternion' multiplication, e.g., (1,2,3)*(1,2,3)= 1,2,3 + 6,2,4 + 6,9,3 = (13,13,10), then -f(t^n) gives the sequence.
Numbers in the sequence are alternatively products of squares or five times a product of squares.
If f(t) is the sum of a+b+c then a(n)=2^(n+1). - Robert G. Wilson v, May 16 2006
FORMULA
t = (-1/p, 1, p). f(t) is the product 1/p*1*p. For t1 = (a, b, c) and t2 = (x, y, z), t1 - t2 = a(x, y, z) + b(z, x, y) + c(y, z, x) = (ax+bz+cy, ay+bx+cz, az+by+cx). -f(t^n) = the sequence.
G.f.: x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)). [Joerg Arndt, Aug 03 2013]
From G. C. Greubel, Sep 21 2020: (Start)
a(n) = 2^(3*n+1) * (1 - (-1)^n * T_{n}(11/16))/27, where T_{n}(x) is the Chebyshev polynomial.
a(n) = -3*a(n-1) + 24*a(n-2) + 512*a(n-3). (End)
EXAMPLE
t = (-0.618...,1,1.618...); t^2 = (3.618...,1.381...,-1). Hence -f(t^2) = 5
MATHEMATICA
s = {-1/GoldenRatio, 1, GoldenRatio}; trit[lst_] := Block[{a, b, c, d, e, f}, {a, b, c} = lst[[1]]; {d, e, f} = lst[[2]]; {{a, b, c}, FullSimplify[{a*d + b*f + c*e, a*e + b*d + c*f, a*f + b*e + c*d}]}]; f[n_] := FullSimplify[ -Times @@ Nest[trit, {s, s}, n][[2]]]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, May 16 2006 *)
CoefficientList[Series[(1 + 8 x) / ((1 - 8 x) (1 + 11 x + 64 x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 04 2013 *)
LinearRecurrence[{-3, 24, 512}, {1, 5, 9}, 20] (* Harvey P. Dale, Feb 28 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Russell Walsmith, Aug 30 2005
EXTENSIONS
More terms from Robert G. Wilson v, May 16 2006
New name using g.f. from Joerg Arndt, Sep 20 2020
STATUS
approved