req proof regarding A196

[Max A.]

On 8/22/06, Joseph Biberstine <jrbibers at indiana.edu> wrote:
>         The following seems to hold for A196 ("Integer part of square root of
> n. Or, number of squares <= n. Or, n appears 2n+1 times.").
>
>         "For n>0, let n = a_0 = 1 + a_1 + ... + a_k with a_(j+1) = a_j -
> floor(sqrt(a_j)).  a(n) is the number of runs in {a_1, ..., a_k}."
>
>         Can anyone prove this?

Something is wrong with the statement.
For example,
n = 10
a_0 = n = 10
a_1 = 10-floor(sqrt(10)) = 7
a_2 = 7-floor(sqrt(7)) = 5
...
But there is no way to represent n as 1 + a_1 + ... + a_k since
1 + a_1 = 8 < n
1 + a_1 + a_2 = 13 > n

Max