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A145051
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Numerator of the first convergent to the sqrt(n) using the recursion x = (n/x+x)/2.
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8
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1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34, 69, 35, 71, 36, 73, 37, 75
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| This is the same as A026741 without the first 2 terms in A026741. The link describes the experimental derivation of the generating function.
Contribution from Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), May 28 2010: (Start)
Numerators of arithmetic means of the first n positive integers for n >= 1.
See A040001 - denominators of arithmetic means of the first n positive integers.
a(n) =A026741(n+1) = A000217(n) * A040001(n) / n. (End)
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LINKS
| Cino Hilliard, Roots by Recursion
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FORMULA
| Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 22 2009: (Start)
G.f.: (1+3x-x^3)/(1-x^2)^2.
a(n)=(n+2)(3-(-1)^n)/4; a(n)=sum{k=0..n, if(k=floor(n/2) or k=floor((n+1)/2),1,0)*(k+1)}. (End)
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EXAMPLE
| n=1;x=1;x=(1/1+1)/2 = 1/1 n=2;x=1;x=(2/1+1)/2 = 3/1 n=3;x=1;x=(3/1+1)/2 = 2/1
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MATHEMATICA
| lst={}; Do[a=n^2+n; b=n^2-n; c=a/b; AppendTo[lst, Denominator[c]], {n, 2, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 20 2009]
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PROG
| (PARI) g(n, p) = x=1; for(j=1, p, x=(n/x+x)/2; if(j==1, print1(numerator(x)", ")))
for(k=1, 100, g(k, 1))
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CROSSREFS
| Cf. A026741
Sequence in context: A194748 A030640 A176447 * A026741 A105658 A083242
Adjacent sequences: A145048 A145049 A145050 * A145052 A145053 A145054
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KEYWORD
| frac,nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)hotmail.com), Sep 30 2008
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