Sequential sums $1+2+\cdots+N$ that are squares

Question

While playing with sums $S_n = 1+\cdots+n$ of integers, I have just come across some "mathematical magic" I have no explanation and no proof for.

Maybe you can give me some comments on this:

I had the computer calculating which Sn are squares, and it came up with the following list:

Table

row $N$ sum($1+\cdots+N$) M (square root of sum)

r=1 N=1 sum=1 M=1

r=2 N=8 sum=36 M=6

r=3 N=49 sum=1225 M=35

r=4 N=288 sum=41616 M=204

r=5 N=1681 sum=1413721 M=1189

r=6 N=9800 sum=48024900 M=6930

Of course we have $1+\cdots+N = \frac{N(N+1)}{2}$, but this gives no indication for which N the sum $1+\cdots+N$ is a square.

Can you guess how in this table we can calculate the entries in row 2 from the entries in row 1? Or the entries in row 3 from the entries in row 2? Or the entries in row 4 from the entries in row 3? Or the entries in row 5 from the entries in row 4?

I looked at the above table and made some strange observations:

1. The value of the next M can be easily calculated from the previous entries: Take the M from the previous row, multiply by 6 and subtract the M from two rows higher up. $M(r) = 6*M(r-1)–M(r-2)$ How is this possible?

   The S(r) we calculate as $S(r) = M(r)^2$. Note that we do not know whether this newly constructed number $S_r$ is in fact of the type $1+\cdots+k$ for some $k$.

2. The value of the next N can be calculated as N(r) = Floor($M(r)*\sqrt 2$), where Floor means “rounding down to the next lower integer“. Somewhat surprising, $S(r)$ is the sum $1+\cdots+N(r)$ !

3. It looks as if outside the entries in the above table there are no other cases. With other words, the method $M(r) = 6*M(r-1)–M(r-2)$ seems to generate ALL solutions n where the sum $1+\cdots+n$ is a square.

Problems:

Is there a proof for any of the three observations? Do observations 1 and 2 really work for the infinite number of rows in this table? Is there an infinite number of rows in the first place?

Puzzled, Karl

Answer 1

I see. Nobody answered this the way I would have... Taking $u = 2 n+1,$ we are solving $$ u^2 - 8 m^2 = 1. $$ A beginning solution is $(3,1).$ Given a solution $(u,m),$ we get a new one $$ (3 u + 8 m, u + 3 m). $$ Then $n = (u-1)/2$ for each pair.

So, with $n^2 + n = 2 m^2$ and $u = 2 n + 1,$ we get triples $$ (n,u,m) $$ $$ (1,3,1) $$ $$ (8,17,6) $$ $$ (49,99,35) $$ $$ (288,577,204) $$ $$ (1681,3363,1189) $$ $$ (9800,19601,6930) $$ $$ (57121,114243,40391) $$ $$ (332928,665857,235416) $$ $$ (1940449,3880899,1372105), $$

With my letters, each is a similar sequence, let us use $r$ for "row," $$ m_1 = 1, m_2 = 6, \; \; m_{r+2} = 6m_{r+1} - m_r, $$ $$ u_1 = 3, u_2 = 17, \; \; u_{r+2} = 6u_{r+1} - u_r, $$ $$ n_1 = 1, n_2 = 8, \; \; n_{r+2} = 6n_{r+1} - n_r + 2. $$

Answer 2

This problem shows up often when working with Pythagorean triangles with consecutive sides. (3,4,5), (20,21,29) etc.

I will answer half the question and the other half is similar. $$\frac{N (N+1)}{2}$$ will be perfect square only when $N$ is square and $(N+1)/2$ is a square or $N/2$ is a square and $N+1$ is a square (this is true since $N$ and $N+1$ are co-prime).

Now consider the first case. $N$ is necessarily odd; so $$ N = (2k+1)^2 = 4k^2+4k+1$$ and $$ \frac{N+1}{2} = 2k^2 + 2k + 1 = k^2 + (k+1)^2$$ Hence we need $$ k^2 + (k+1)^2 = m^2$$ The solution is just Pythagorean triangle with consecutive sides. Refer to any elementary number theory book for finding these.