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A142238
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Numerators of continued fraction convergents to sqrt(3/2).
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10
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1, 5, 11, 49, 109, 485, 1079, 4801, 10681, 47525, 105731, 470449, 1046629, 4656965, 10360559, 46099201, 102558961, 456335045, 1015229051, 4517251249, 10049731549, 44716177445, 99482086439, 442644523201, 984771132841, 4381729054565, 9748229241971
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OFFSET
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0,2
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COMMENTS
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In general, denominators, a(k,n) and numerators, b(k,n), of continued
fraction convergents to sqrt((k+1)/k) may be found as follows:
a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)
and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2)
and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
For example, the convergents to sqrt(3/2) start 1/1, 5/4, 11/9,
49/40, 109/89.
In general, if a(k,n) and b(k,n) are the denominators and numerators,
respectively, of continued fraction convergents to sqrt((k+1)/k)
as defined above, then
k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and
b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n);
for example, if k=2 and n=3, then b(2,n)=a(n) and
2*a(2,6)^2-a(2,5)*a(2,7)=2*881^2-396*3920=2;
2*a(2,4)*a(2,6)-a(2,5)^2=2*89*881-396^2=2;
b(2,5)*b(2,7)-2*b(2,6)^2=485*4801-2*1079^2=3;
b(2,5)^2-2*b(2,4)*b(2,6)=485^2-2*109*1079=3.
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LINKS
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FORMULA
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G.f.'s for numerators and denominators are -(1+5*x+x^2-x^3)/(-1-x^4+10*x^2) and -(1+4*x-x^2)/(-1-x^4+10*x^2).
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EXAMPLE
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The initial convergents are 1, 5/4, 11/9, 49/40, 109/89, 485/396, 1079/881, 4801/3920, 10681/8721, 47525/38804, 105731/86329, ...
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MAPLE
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with(numtheory): cf := cfrac (sqrt(3)/sqrt(2), 100): [seq(nthnumer(cf, i), i=0..50)]; [seq(nthdenom(cf, i), i=0..50)]; [seq(nthconver(cf, i), i=0..50)];
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MATHEMATICA
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Numerator[Convergents[Sqrt[3/2], 30]] (* Bruno Berselli, Nov 11 2013 *)
LinearRecurrence[{0, 10, 0, -1}, {1, 5, 11, 49}, 30] (* Harvey P. Dale, Dec 30 2017 *)
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PROG
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(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 10, 0]^n*[1; 5; 11; 49])[1, 1] \\ Charles R Greathouse IV, Jun 21 2015
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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N. J. A. Sloane, Oct 05 2008, following a suggestion from Rob Miller (rmiller(AT)AmtechSoftware.net)
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STATUS
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approved
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