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%I #33 Feb 28 2024 19:24:41
%S 1,5,9,605,961,16245,284089,29645,15046641,101025125,73222249,
%T 9908816445,23755748641,204034140245,5031349566489,1965713970605,
%U 219320727489361,1965265930868805,374345220088009,158335559155140125
%N Expansion of x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)).
%C 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.
%C Numbers in the sequence are alternatively products of squares or five times a product of squares.
%C If f(t) is the sum of a+b+c then a(n)=2^(n+1). - _Robert G. Wilson v_, May 16 2006
%H Vincenzo Librandi, <a href="/A112259/b112259.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,24,512).
%F 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.
%F G.f.: x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)). [_Joerg Arndt_, Aug 03 2013]
%F From _G. C. Greubel_, Sep 21 2020: (Start)
%F a(n) = 2^(3*n+1) * (1 - (-1)^n * T_{n}(11/16))/27, where T_{n}(x) is the Chebyshev polynomial.
%F a(n) = -3*a(n-1) + 24*a(n-2) + 512*a(n-3). (End)
%e t = (-0.618...,1,1.618...); t^2 = (3.618...,1.381...,-1). Hence -f(t^2) = 5
%t 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 *)
%t CoefficientList[Series[(1 + 8 x) / ((1 - 8 x) (1 + 11 x + 64 x^2)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 04 2013 *)
%t LinearRecurrence[{-3,24,512},{1,5,9},20] (* _Harvey P. Dale_, Feb 28 2024 *)
%Y Cf. A001622, A112260, A112261.
%K nonn,easy
%O 1,2
%A _Russell Walsmith_, Aug 30 2005
%E More terms from _Robert G. Wilson v_, May 16 2006
%E New name using g.f. from _Joerg Arndt_, Sep 20 2020
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