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Search: id:A112259
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| A112259 |
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Let p = the golden mean = (1+5^0.5)/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. |
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3
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| 1, 5, 9, 605, 961, 16245, 284089, 29645, 15046641, 101025125, 73222249, 9908816445, 23755748641, 204034140245, 5031349566489, 1965713970605, 219320727489361, 1965265930868805, 374345220088009, 158335559155140125
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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.
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LINKS
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Russell Walsmith, Triternions.
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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.
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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 (rgwv(at)rgwv.com), May 16 2006
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CROSSREFS
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Cf. A112260, A112261.
Sequence in context: A109076 A101683 A098135 this_sequence A099731 A091306 A073048
Adjacent sequences: A112256 A112257 A112258 this_sequence A112260 A112261 A112262
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KEYWORD
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nonn
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AUTHOR
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Russell Walsmith (russw(AT)lycos.com), Aug 30 2005
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(at)rgwv.com), May 16 2006
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