A298737 Numerators of successive rational approximations converging to 2*Pi from above for n >= 1, with a(-1) = 0 and a(0) = 1.

0, 1, 7, 13, 19, 44, 377, 710, 104703, 208696, 312689, 2292816, 6565759, 10838702, 90982559, 171126416, 251270273, 331414130, 411557987, 2549491779, 14885392687, 56992078969, 99098765251, 141205451533, 183312137815, 225418824097, 267525510379, 309632196661, 351738882943, 393845569225, 435952255507

Offset

-1,3

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..29.

Formula

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)).

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) = 7, a(2) = 13, a(3) = 19.

Crossrefs

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

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

Sequence in context: A358322 A181938 A073648 * A109558 A108106 A231506

Adjacent sequences: A298734 A298735 A298736 * A298738 A298739 A298740

Keyword

frac,nonn

Author

Allan C. Wechsler, Mar 18 2018

Extensions

Offset corrected by Altug Alkan, Mar 19 2018

More terms from Altug Alkan and N. J. A. Sloane (independently), Mar 19 2018

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

Status

approved