A263295 Denominators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...).

3, 5, 33, 111, 933, 4913, 50585, 364717, 4460647, 40920133, 580574377, 6472209467, 104581586665, 1373040648769, 24902871413201, 376386726269561, 7573128424949291, 129519388933667493, 2863373356440803473, 54670305859684290279, 1317404009250178503245

Offset: 1

Author

Mohamed Sabba, Nov 20 2015

Keywords

• nonn
• frac

Comments

As n->inf, A262957(n)/a(n) converges to 0.57663338973018...; this number has a surprisingly elegant standard continued fraction representation: [0; 1, 1, 2, 1, 3, 4, 1, 5, 6, 1, 7, 8, ...].

Examples

			(1-1/(2+1)) = 2/3, so a(1) = 3;
(1-1/(2+1/(3-1))) = 3/5, so a(2) = 5;
(1-1/(2+1/(3-1/(4+1)))) = 19/33, so a(3) = 33;
(1-1/(2+1/(3-1/(4+1/(5-1))))) = 64/111, so a(4) = 111.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..448

Crossrefs

Same principle as A244279 and A244280 - except here we begin with subtraction rather than addition.

Cf. A262957 (numerators).

Programs

PARI

a(n) = if(n%2==0, s=-1, s=1); t=1; while(n>0, t=n+1+s/t; n--; s=-s); denominator(t=1/t)
vector(30, n, a(n)) \\ corrected by Mohammed Sabba, Dec 22 2015

Extensions

More terms from Mohamed Sabba, Dec 22 2015