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A112259
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Expansion of x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)).
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4
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1, 5, 9, 605, 961, 16245, 284089, 29645, 15046641, 101025125, 73222249, 9908816445, 23755748641, 204034140245, 5031349566489, 1965713970605, 219320727489361, 1965265930868805, 374345220088009, 158335559155140125
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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]
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)
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EXAMPLE
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t = (-0.618...,1,1.618...); t^2 = (3.618...,1.381...,-1). Hence -f(t^2) = 5
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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