Anonymous at Sat, 29 Mar 2025 13:53:28 UTC No. 16631782

>>16631525
The limit of the sequence a_1, a_3, a_5, ... is sqrt(2).
The limit of the sequence a_2, a_4, a_6, ... is 1/sqrt(2).

Anonymous at Sat, 29 Mar 2025 14:02:18 UTC No. 16631788

>>16631525
a_1 > a_3 > a_5 > a_7 > ... > sqrt(2)
a_2 < a_4 < a_6 < a_8 < ... < 1/sqrt(2)

Anonymous at Sat, 29 Mar 2025 18:35:12 UTC No. 16631963

>>16631525
a_5

Anonymous at Sat, 29 Mar 2025 23:00:06 UTC No. 16632146

>>16631525
i don't understand how the sequence is meant to uniquely continue. i'm trying to write down a_4, and it's unclear what to write.

a_2 = (4/3) / a_1
a_3 = [(8/7) / (6/5)] / a_2

so is it

a_4 = [ (16/15) / (14/13) / (12/11) / (10/9) ] / a_3

but i'm unsure where the parenthesis go?

Anonymous at Sat, 29 Mar 2025 23:14:55 UTC No. 16632155

>>16632146
okay i see the pattern now.

a_1 = 2
a_2 = (4/3)/a_1
a_3 = [8*5 / (7*6)] / a_2
a_4 = [16*13*11*10 / (15*14*12*9) ] / a_3
that's for even a_n. so it's a skipping, growing factorial. odd a_n is a little different

Anonymous at Sun, 30 Mar 2025 02:32:33 UTC No. 16632226

z_1 = 1/2
z_2 = (1/2)/(3/4) = 2/3
z_3 = [(1/2)/(3/4)]/[(5/6)/(7/8)] = 7/10
z_4 = ([(1/2)/(3/4)]/[(5/6)/(7/8)])/([(9/10)/(11/12)]/[(13/14)/(15/16)]) = 286/405
...
The foregoing sequence doesn't oscillate.
And its limit is sqrt(1/2).

Anonymous at Sun, 30 Mar 2025 02:39:11 UTC No. 16632228

>>16631525
Is this a biology, or cell division, sequence?

Anonymous at Sun, 30 Mar 2025 03:52:43 UTC No. 16632257

it seems to oscillate irregularly, i dont think it converges

Anonymous at Sun, 30 Mar 2025 04:17:39 UTC No. 16632260

d(n) = number of 1's in binary representation of n

a(2n+1) = b(n), a(2n) = 1/b(n) where b(n) = product of k^(-1)^d(n-1) from 1 to 2^n

b(n) -> sqrt(2) clearly but I hate problems with digit representations so someone else do it

Anonymous at Sun, 30 Mar 2025 04:36:45 UTC No. 16632266

>>16632257
>it seems to oscillate irregularly
Didn't you read the 2nd and 3rd posts?

Anonymous at Mon, 31 Mar 2025 08:37:12 UTC No. 16633553

bamp

Anonymous at Mon, 31 Mar 2025 08:42:41 UTC No. 16633558

>>16631525
That's just a simplicial adjunct to Collatz...