A299923 Denominators of successive rational approximations converging to 2*Pi from above for n >= 1, with a(-1) = -1 and a(0) = 0.

-1, 0, 1, 2, 3, 7, 60, 113, 16664, 33215, 49766, 364913, 1044973, 1725033, 14480324, 27235615, 39990906, 52746197, 65501488, 405764219, 2369083826, 9070571085, 15772058344, 22473545603, 29175032862, 35876520121, 42578007380, 49279494639, 55980981898, 62682469157, 69383956416, 76085443675, 82786930934, 89488418193

Offset

-1,4

Comments

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

Links

Table of n, a(n) for n=-1..32.

Formula

Set a(-1) = -1; a(0) = 0; 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)).

Example

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) = 1, a(2) = 2, a(3) = 3. [Corrected by Altug Alkan, Mar 19 2018]

Crossrefs

Cf. A298737.

Sequence in context: A159611 A156585 A354744 * A337189 A087358 A255357

Adjacent sequences: A299920 A299921 A299922 * A299924 A299925 A299926

Keyword

frac,sign

Author

Allan C. Wechsler, Mar 18 2018

Extensions

More terms from N. J. A. Sloane, Mar 19 2018

a(-1) = -1 and a(0) = 0 prepended by Altug Alkan, Mar 26 2018

Status

approved