My formula for sum of consecutive squares series?

Question

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers.

For exmaple, this first number in the series results in: $3^2 + 4^2 = 5^2$.

The second results in: $10^2 + 11^2 + 12^2 = 13^2 + 14^2$.

The thrid: $21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2$. etc.

The formula for the series is:

By using; $1^2 + 2^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6}$ it is easy to prove that both side are the same.

Have I found something new? Useful?

I have a feeling that this is the only series with this specific property (consecutive sum of squares equal continuation of consecutive sum of squares), but I am unsure how to prove that.

Any comments would be appreciated.

Answer 1

$$\sum_{j=60}^{110} j^2 = \sum_{j=111}^{135} j^2$$ is a solution with $51$ terms on the left and $25$ terms on the right side.

So, the conjecture that there are always $x+1$ and $x$ terms, is false.

Answer 2

Well, $3^2+4^2=5^2$ has been around for quite a while; the rest is less ancient, but hardly new anyway.

Indeed, if we look up the OEIS for your sequence, the very first comment says:

Note that when starting from $a_n^2$, equality holds between series of first $n+1$ and next $n$ consecutive squares: $$a_n^2+(a_n+1)^2+\dots+(a_n+n)^2 = (a_n+n+1)^2+(a_n+n+2)^2+\dots(a_n+2n)^2;$$ e.g., $10^2+11^2+12^2 = 13^2+14^2$.

(I took the liberty of formatting the quote above. OEIS is run by seasoned math geeks who prefer to read their LaTeX code raw.)

Answer 3

Here are the examples of $\sum_{j=a}^{b} j^2 = \sum_{j=b+1}^{c} j^2$ where the limits are less than $1000$.

There are examples that do not follow your pattern. But I don't know whether there are other counterexamples beyond $1000$.

     sum        a       b       c       L       R
      25        3       4       5       2       1       true
     365        10      12      14      3       2       true
    2030        21      24      27      4       3       true
    7230        36      40      44      5       4       true
   11900        18      34      42      17      8       false
   19005        4       38      48      35      10      false
   19855        55      60      65      6       5       true
   42419        12      50      63      39      13      false
   45955        78      84      90      7       6       true
   94220        105     112     119     8       7       true
  176460        136     144     152     9       8       true
  308085        171     180     189     10      9       true
  379525        60      110     135     51      25      false
  508585        210     220     230     11      10      true
  802010        253     264     275     12      11      true
  963295        16      142     179     127     37      false
 1217450        300     312     324     13      12      true
 1254539        67      159     198     93      39      false
 1789515        351     364     377     14      13      true
 2558815        406     420     434     15      14      true
 3572440        465     480     495     16      15      true
 4884440        528     544     560     17      16      true
 6556305        595     612     629     18      17      true
 8657445        666     684     702     19      18      true
11265670        741     760     779     20      19      true
14467670        820     840     860     21      20      true
18359495        903     924     945     22      21      true