TITLE: Can every natural number be written as a sum of signed odd squares?
QUESTION [8 upvotes]: Let $c_n \in \{ -1,1\}$. Here, it is stated that every natural number may be written as
$$\sum c_kk^2$$
Where $k$ runs from $1$ to some finite number. I am wondering whether every natural number $n$ can be written as follows:
$$n = \sum c_n(2k-1)^2.$$
In other words, 

can every natural number be written as the sum of the first so-and-so signed odd squares?

Obviously, $1=1^2$. However, even to find such a writing of $2$, I needed eight squares: 
$$2=1+9+25-49+81-121-169+225$$
And could not find one for $3$. Any insight would be appreciated.

REPLY [2 votes]: Shortest representations of small positive integers.
\begin{align*}
1 &= 1  \\
2 &= 1 + 9 + 25 - 49 + 81 - 121 - 169 + 225  \\
3 &= 1 + 9 + 25 + 49 - 81  \\
4 &= -1-9-25-49+81-121-169+225-289+361  \\
5 &= 1+9+25+49-81+121+169+225+289-361-441  \\
6 &= -1-9+25-49-81+121  \\
7 &= 1+9+25-49+81+121+169+225+289+361+441+529-625-729-841  \\
  &= 1+9+25+49+81+121+169+225+289-361+441-529-625-729+841 \\
8 &= -1+9  \\
9 &= -1-9-25-49-81+121-169-225-289-361-441-529+625+729+841+961-1089  \\
  &= -1-9+25-49-81-121-169-225-289+361-441-529-625-729+841+961+1089  \\
10 &= 1 + 9  \\
11 &= -1-9+25-49-81-121-169-225+289-361-441+529+625  \\
12 &= -1-9-25-49-81+121-169+225  \\
13 &= -1-9-25-49+81+121-169-225+289  \\
14 &= -1-9-25+49  \\
15 &= -1-9+25  \\
16 &= 1-9-25+49  \\
17 &= 1-9+25  \\
18 &= 1+9-25+49-81+121+169-225  \\
19 &= 1+9+25-49+81+121-169  \\
20 &= -1-9+25-49-81-121-169-225+289+361  \\
21 &= 1+9+25+49+81+121+169-225-289-361+441  \\
22 &= -1-9+25-49-81+121+169-225-289+361  \\
  &= -1+9-25-49+81-121-169+225-289+361  \\
  &= 1-9+25-49-81-121-169-225+289+361  \\
23 &= 1+9+25+49+81+121+169-225+289+361-441+529+625-729-841  \\
24 &= -1-9+25+49+81-121  \\
  &= -1+9+25-49-81+121  \\
25 &= -1-9-25-49-81-121-169-225-289-361+441+529+625+729-841+961-1089  \\
  &= -1-9-25-49-81-121-169+225-289-361-441+529-625+729+841+961-1089  \\
  &= -1-9-25-49-81-121-169-225+289-361+441-529+625-729+841-961+1089  \\
  &= 1-9-25-49-81-121-169-225-289-361-441-529+625+729+841-961+1089  \\
  &= -1-9-25-49-81-121+169-225-289-361+441-529+625-729-841+961+1089  \\
  &= -1-9-25-49+81-121-169-225-289-361-441+529+625-729-841+961+1089  \\
  &= -1-9-25-49-81-121+169+225-289-361-441-529-625-729+841+961+1089  \\
26 &= 1-9+25+49+81-121  \\
  &= 1+9+25-49-81+121  \\
27 &= -1-9-25-49-81-121+169-225+289-361+441  \\
28 &= -1+9+25+49+81+121+169+225-289-361  \\
  &= 1+9+25+49+81-121-169+225+289-361  \\
29 &= -1-9-25-49+81-121+169-225+289+361-441  \\
  &= -1-9+25-49-81-121-169+225+289+361-441  \\
  &= -1-9-25+49-81+121-169-225+289-361-441  \\
  &= 1-9-25-49-81-121+169-225+289-361+441  \\
30 &= -1-9-25+49+81-121-169+225  \\
  &= -1+9-25-49-81+121-169+225  \\
31 &= -1-9+25+49-81-121+169  \\
32 &= -1+9-25+49  \\
33 &= -1+9+25  \\
34 &= 1+9-25+49  \\
35 &= 1+9+25  \\
36 &= -1-9-25-49-81-121+169+225+289-361  \\
37 &= -1+9+25+49+81+121+169=225-289+361+441-529-625  \\
  &= 1+9+25+49+81-121-169+225+289+361+441-529-625  \\
  &= 1+9+25+49+81+121-169+225-289-361+441+529-625  \\
38 &= -1-9-25+49-81+121-169+225+289-361  \\
  &= 1-9-25-49-81-121+169+225+289-361  \\
  &= -1+9+25-49-81-121-169-225+289+361  \\
39 &= 1+9+25+49+81+121+169+225-289+361+441-529-625  \\
40 &= -1-9+25-49+81+121+169-225+289-361  \\
  &= 1-9-25+49-81+121-169+225+289-361  \\
  &= -1-9+25+49+81-121+169-225-289+361 \\
  &= -1+9+25-49-81+121+169-225-289+361  \\
  &= 1+9+25-49-81-121-169-225+289+361
\end{align*}