# [seqfan] Re: Number of ways to write n^2 as the sum of n odd numbers

Max Alekseyev maxale at gmail.com
Fri Sep 25 20:40:38 CEST 2009

```It can be expressed as
A008284((n^2+n)/2,n) = A008284(A000217(n),n).

Regards,
Max

On Fri, Sep 25, 2009 at 1:11 PM, David Radcliffe <dradcliffe at gmail.com> wrote:
> Greetings Sequence Fans,
>
> Everybody knows that the nth square is equal to the sum of the first n
> odd numbers.
> It might be interesting to calculate the number of ways to write n^2
> as the sum of n
> odd numbers, disregarding order.
>
> For example, 9 can be written as a sum of three odd numbers in 3 ways: 1+1+7,
> 1+3+5, and 3+3+3.
>
> I wrote Maple program to compute the first few terms of this sequence. It does
> not match any sequence in the OEIS. It would be interesting to know the
> asymptotic behavior or a simpler formula for this sequence.
>
> The first 20 terms are:
> 1, 1, 3, 9, 30, 110, 436, 1801, 7657, 33401, 148847, 674585, 3100410, 14422567,
> 67792847, 321546251, 1537241148, 7400926549, 35854579015, 174677578889.
>
> My Maple code is listed below.
>
> f := proc (n, k) option remember;
>   if n = 0 and k = 0 then return 1 end if;
>   if n <= 0 or n < k then return 0 end if;
>   if `mod`(n+k, 2) = 1 then return 0 end if;
>   if k = 1 then return 1 end if;
>   return f(n-1, k-1) + f(n-2*k, k)
> end proc;
>
> seq(f(k,k^2), k=1..20);
>
> --