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A081459
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Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with r = 2 and applying the mapping to each new (reduced) rational number gives 2, 9/4, 161/72, 51841/23184, ..., tending to N^(1/2). Sequence gives values of the numerators.
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3
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2, 9, 161, 51841, 5374978561, 57780789062419261441, 6677239169351578707225356193679818792961, 89171045849445921581733341920411050611581102638589828325078491812417901966295041
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6 etc.
The sequence satisfies the Pell equation a(n+1)^2 - 5*A081460(n+1)^2 = 1. - Vincenzo Librandi, Dec 20 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..11
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FORMULA
| a(n) = a(n-1)^2+5*A081460(n-1)^2. - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003
a(n) = (1/2)*(((4+2*sqrt(5))/2)^(2^(n-1))+((4-2*sqrt(5))/2)^(2^(n-1))). a(n+1) = 2*a(n)^2-1 for n>1. - Artur Jasinski (grafix(AT)csl.pl), Oct 12 2008
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MATHEMATICA
| Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 12 2008: (Start)
k = 4; Table[Simplify[Expand[(1/2) (((k + Sqrt[k^2 + 4])/2)^(2^(n - 1)) + ((k - Sqrt[k^2 + 4])/2)^(2^(n - 1)))]], {n, 1, 6}]
or
aa = {}; k = 9; Do[AppendTo[aa, k]; k = 2 k^2 - 1, {n, 1, 5}]; aa (*Artur Jasinski*) (End)
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PROG
| (PARI) {r=2; N=5; for(n=1, 8, a=numerator(r); b=denominator(r); print1(a, ", "); r=(1/2)*(r + N/r) )}
(MAGMA) m:=8; f:=[ n eq 1 select 2 else (Self(n-1)+5/Self(n-1))/2: n in [1..m] ]; [ Numerator(f[n]): n in [1..m] ]; // Bruno Berselli, Dec 20 2011
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CROSSREFS
| Cf. A000129, A001333, A081460.
Sequence in context: A117116 A133468 A182948 * A038843 A053294 A199695
Adjacent sequences: A081456 A081457 A081458 * A081460 A081461 A081462
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KEYWORD
| nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
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EXTENSIONS
| Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de) and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
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