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A142238 Numerators of continued fraction convergents to sqrt(3/2). 10

%I

%S 1,5,11,49,109,485,1079,4801,10681,47525,105731,470449,1046629,

%T 4656965,10360559,46099201,102558961,456335045,1015229051,4517251249,

%U 10049731549,44716177445,99482086439,442644523201,984771132841,4381729054565,9748229241971

%N Numerators of continued fraction convergents to sqrt(3/2).

%C From _Charlie Marion_, Jan 07 2009: (Start)

%C In general, denominators, a(k,n) and numerators, b(k,n), of continued

%C fraction convergents to sqrt((k+1)/k) may be found as follows:

%C a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)

%C and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);

%C b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2)

%C and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).

%C For example, the convergents to sqrt(3/2) start 1/1, 5/4, 11/9,

%C 49/40, 109/89.

%C In general, if a(k,n) and b(k,n) are the denominators and numerators,

%C respectively, of continued fraction convergents to sqrt((k+1)/k)

%C as defined above, then

%C 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

%C 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);

%C for example, if k=2 and n=3, then b(2,n)=a(n) and

%C 2*a(2,6)^2-a(2,5)*a(2,7)=2*881^2-396*3920=2;

%C 2*a(2,4)*a(2,6)-a(2,5)^2=2*89*881-396^2=2;

%C b(2,5)*b(2,7)-2*b(2,6)^2=485*4801-2*1079^2=3;

%C b(2,5)^2-2*b(2,4)*b(2,6)=485^2-2*109*1079=3.

%C Cf. A000129, A001333, A142239, A153313-153318. (End)

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).

%F 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).

%F A142238(2n) = A041006(2n)/2 = A054320(n), A142238(2n-1) = A041006(2n-1) = A041038(2n-1) = A001079(n). - _M. F. Hasler_, Feb 14 2009

%e 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, ...

%p 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)];

%t Numerator[Convergents[Sqrt[3/2], 30]] (* _Bruno Berselli_, Nov 11 2013 *)

%t LinearRecurrence[{0,10,0,-1},{1,5,11,49},30] (* _Harvey P. Dale_, Dec 30 2017 *)

%o (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

%Y Cf. A115754, A142239.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_, Oct 05 2008, following a suggestion from Rob Miller (rmiller(AT)AmtechSoftware.net)

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Last modified February 19 18:30 EST 2019. Contains 320327 sequences. (Running on oeis4.)