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A115730
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Dihedral D2 elliptical invariant transform on A000045: a[n+1]/a[n]= 1+Phi=1+(1+Sqrt[5])/2.
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5
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0, 0, 1, 2, 6, 16, 42, 110, 289, 756, 1980, 5184, 13572, 35532, 93025, 243542, 637602, 1669264, 4370190, 11441306, 29953729, 78419880, 205305912, 537497856, 1407187656, 3684065112, 9645007681, 25250957930, 66107866110
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| A D2 elliptical invariant transform gives a ratio of 1+Phi.
The a(n+1) represent the Ca2 and Ze4 sums of the Golden Triangle A180662. Furthermore the a(3*n) represent the Ze1 (terms doubled) and Ca3 sums of the Golden triangle. See A180662 for more information about these and other triangle sums.
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FORMULA
| b[n]=A000045[n] g[x]=(x^2-1)^2/(-4*x^2): D2 dihedral elliptical invariant function a(n) = -Floor[g[b[n+1]]
a(n)=a(n-3)+A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1. The A001654(n) are the golden rectangle numbers.
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MAPLE
| nmax:=31: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):=a(n-3) + A001654(n-1) od: seq(a(n), n=0..nmax);
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MATHEMATICA
| F[0] = 0; F[1] = 1; F[n_] := F[n] = F[n - 1] + F[n - 2] g[x_] = (x^2 - 1)^2/(-4*x^2) a = Table[ -Floor[g[F[n]]], {n, 1, 32}] Table[N[a[[n + 1]]/a[[n]]], {n, 1, Length[a] - 1}]
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CROSSREFS
| Cf. A000045, A079962, A064831, A180664, A180665, A180666.
Sequence in context: A156664 A025169 A111282 * A191694 A027994 A027068
Adjacent sequences: A115727 A115728 A115729 * A115731 A115732 A115733
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 13 2006
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EXTENSIONS
| Corrected and information added by Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010
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