A224365 a(n) = A063674(n+1) - A063674(n).

10, 3, 3, 3, 157, 22, 22, 22, 22, 22, 22, 22, 22, 51808, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355

Offset: 1

Author

Paul Curtz, Apr 09 2013

Keywords

• nonn
• less
• frac

Comments

The repeated terms (3, 22, 355, 5419351, ... from A063674) are the numerators of fractions (3/1, 22/7, 355/113, 5419351/1725033, ...) leading to Pi.

Zu Chongzhi (5th century) discovered 22/7 and 355/113. Adriaan Anthonisz Metius rediscovered 355/113 in 1585.

First differences of A063673 give the denominators: 3, 1, 1, 1, 50, 7, 7, 7, 7, 7, 7, 7, 7, 16489, 113, 113, ... .

Hence 10/3, 157/50, 51808/16489, ... .

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..168

Crossrefs

Cf. A063673, A063674.

Cf. A046947, A072398, A002485. A132049, A003077, A068028, A068079.

Programs

Mathematica

A224365 = Reap[ For[ delta0 = 1; d = 1, d < 10^5, d++, delta = Abs[Pi - Round[Pi*d]/d]; If[ delta < delta0, Sow[ Round[Pi*d]]; delta0 = delta]]][[2, 1]] // Differences (* Jean-François Alcover, Apr 10 2013 *)

Formula

a(n) = A063674(n+1) - A063674(n).