A298737 - OEIS

%I #126 Mar 26 2018 20:57:57

%S 0,1,7,13,19,44,377,710,104703,208696,312689,2292816,6565759,10838702,

%T 90982559,171126416,251270273,331414130,411557987,2549491779,

%U 14885392687,56992078969,99098765251,141205451533,183312137815,225418824097,267525510379,309632196661,351738882943,393845569225,435952255507

%N Numerators of successive rational approximations converging to 2*Pi from above for n >= 1, with a(-1) = 0 and a(0) = 1.

%C Suggested by Henry Baker in a message to the math-fun mailing list, Mar 16 2018.

%F Set a(-1) = 0; a(0) = 1; a(n+1) = c(n) * a(n) - a(n-1), where t(0) = 2*Pi, c(n) = ceiling (t(n)), and t(n+1) = 1/(c(n) - t(n)).

%e The best integer over-estimate of 2*Pi is 7. Between 2*Pi and 7 the rational with the smallest denominator is 13/2. Between 2*Pi and 13/2, the rational with the smallest denominator is 19/3. So a(1) = 7, a(2) = 13, a(3) = 19.

%Y Cf. A046995, a similar sequence of numerators of rationals converging to 2*Pi, the traditional continued fraction convergents.

%Y For the c sequence see A299922, and for the denominators see A299923.

%K frac,nonn

%O -1,3

%A _Allan C. Wechsler_, Mar 18 2018

%E Offset corrected by _Altug Alkan_, Mar 19 2018

%E More terms from _Altug Alkan_ and _N. J. A. Sloane_ (independently), Mar 19 2018

%E a(-1) = 0 prepended by _Altug Alkan_, Mar 26 2018