1,1

sum{k=1 to n} 1/a(k) increases, but is bounded from above (by the product). While product{j=1 to n} 1/(1 +1/a(j)) decreases and is bounded from below (by the sum). The sum and the product then approach the same constant, which is approximately .6037789..., if their difference approaches 0. Does this constant have a closed form in terms of known constants, if the constant exists?

Leroy Quet, Home Page (listed in lieu of email address)

For n >= 2, if x = product{j=1 to n-1} 1/(1 +1/a(j)) and y = sum{k=1 to n-1} 1/a(k), then a(n) = ceiling[(1 + y + sqrt((y-1)^2 + 4x))/(2(x-y))].

sum{k=1 to 2} 1/a(k) = 3/5 and product{j=1 to 2} 1/(1 +1/a(j)) = 20/33. For m = any positive integer <= 264, 3/5 + 1/m is > 20/33/(1 + 1/m). But if m = 265, then 3/5 + 1/m = 32/53 is <= 20/33/(1 + 1/m) = 2650/4389. So a(3) = 265.

Digits := 220 ; A134473 := proc(n) option remember ; local su, mu ; if n =1 then 2; else su := add(1/procname(k), k=1..n-1) ; mu := mul(1/(1+1/procname(j)), j=1..n-1) ; ceil( (1+su+sqrt((su-1)^2+4*mu))/2/(mu-su) ) ; fi; end: seq(A134473(n), n=1..9) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 20 2009]

Cf. A134474, A134475, A134476, A134477.

Sequence in context: A159558 A001528 A088310 this_sequence A005154 A074056 A144288

Adjacent sequences: A134470 A134471 A134472 this_sequence A134474 A134475 A134476

nonn

Leroy Quet Oct 27 2007

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 20 2009