A076539 Numerators a(n) of fractions slowly converging to Pi: let a(1) = 0, b(n) = n - a(n); if (a(n) + 1) / b(n) < Pi, then a(n+1) = a(n) + 1, otherwise a(n+1) = a(n).

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 50, 51, 52, 53, 53, 54, 55

Offset: 1

Author

Robert A. Stump (bee_ess107(AT)msn.com), Oct 18 2002

Keywords

• easy
• frac
• nonn

Comments

a(n) + b(n) = n and as n -> +infinity, a(n)/b(n) converges to Pi. For all n, a(n)/b(n) < Pi.

Examples

			a(7)= 5 so b(7) = 7 - 5 = 2.
a(8) = 6 because (a(7) + 1)/b(7) = 6/2 which is < Pi. So b(8) = 8 - 6 = 2.
a(9) = 6 because (a(8) + 1)/b(8) = 7/2 which is not < Pi.
		

Crossrefs

Cf. A074840, A074065, A060143.

Partial sums of A144609.

Programs

Mathematica

Array[Floor[# Pi/(Pi + 1)] &, 73] (* Michael De Vlieger, Jan 11 2018 *)

Formula

a(1) = 0, b(n) = n - a(n), if (a(n) + 1)/b(n) < Pi, then a(n+1) = a(n) + 1, otherwise a(n+1) = a(n).

a(n) = floor(n*Pi/(Pi+1)). - Vladeta Jovovic, Oct 04 2003