A042936 - OEIS

%I #26 Jun 26 2022 23:36:50

%S 31,32,63,95,158,253,1676,3605,8886,136895,282676,702247,4496158,

%T 5198405,9694563,14892968,24587531,39480499,2472378469,2511858968,

%U 4984237437,7496096405,12480333842,19976430247,132338915324,284654260895,701647437114,10809365817605,22320379072324

%N Numerators of continued fraction convergents to sqrt(1000).

%H Vincenzo Librandi, <a href="/A042936/b042936.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78960998, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

%t Numerator[Convergents[Sqrt[1000], 30]] (* _Harvey P. Dale_, Oct 29 2013 *)

%o From  _M. F. Hasler_, Nov 01 2019: (Start)

%o (PARI) A42936=contfracpnqn(c=contfrac(sqrt(1000)), #c)[1,][^-1] \\ Discards possibly incorrect last term. NB: a(n)=A42936[n+1]. Could be extended using: {A42936=concat(A42936, 78960998*A42936[-18..-1]-A42936[-36..-19])}

%o \\ But terms with arbitrarily large indices can be computed using:

%o A042936(n)={[A42936[n%18+i]|i<-[1, 19]]*([0, -1; 1, 78960998]^(n\18))[,1]} \\ Faster but longer with n=divrem(n,18). (End)

%Y Cf. A042937 (denominators).

%Y Analog for sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042934 (m=999).

%K nonn,frac,easy

%O 0,1

%A _N. J. A. Sloane_