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This generalized Pell equation appears in the solution of problems posed in A001032 (and A001033): numbers n such that the sum of squares of n consecutive (odd) positive integers is a square. This sequence is the union of A001032, A001033 and the number 4, which is not a solution to either problem. When n is a square > 1 and not divisible by 3, then the equation has only a finite number of solutions; otherwise it has an infinite number of solutions.

For an n in this sequence, consider solutions with x>0 and y>n. (For n=4, there will be no such solutions.) If y-n+1 is even, then n is in A001032, the n consecutive positive integers begin with (y-n+1)/2 and the sum of the squares is x/2. If y-n+1 is odd, then the n is in A001033, the n consecutive odd positive integers begin with y-n+1 and the sum of the squares is x. For some n, such as 33, there are solutions y1 and y2 such that y1-n+1 is even and y2-n+1 is odd. In this case, n is in both A001032 and A001033.

The reason that 4 is not in A001033 is that there is no sequence of 4 consecutive positive odd squares that add to a square. However, there is a sequence of 4 consecutive odd integers whose squares add up to a square: (-1)^2 + 1^2 + 3^2 + 5^2 = 6^2. - Thomas Andrews, Feb 22 2011

A001032 consists of those n for which there is a sequence of n consecutive positive squares whose sum is a square. For the associated Pellian equation, see A134419. The necessary congruence conditions described in A274471 apply here:

(defining x||y to mean x|y and x and y/x are coprime)

if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);

if p^e||n with p=5 or 7 (mod 12), then e is even;

if 3^e||(n+1) with e>0, then e is odd;

if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.

In addition, in order that the Pellian equation has solutions of the correct parity, one must have:

if 2^e||n with e>0, then e is odd.

However, these conditions are not sufficient. This sequence consists of the numbers n that satisfy all the congruence conditions but for which there is no sequence of n consecutive positive squares whose sum is a square.

The term 25 is present despite the Pellian equation having a solution with the correct parity, because it leads only to 0^2 + 1^2 + ... + 24^2 = 70^2 and the specification of A001032 requires the squares to be strictly positive. (One may wonder whether this is quite natural, compare A185545.) In every other case the Pellian equation lacks solutions with the right parity. Note however that it may still have solutions with the opposite parity (this can happen only if n=1 mod 8) and so this sequence is not a subsequence of A274471.

A001033 consists of those n for which there is a sequence of n consecutive positive odd squares whose sum is a square. For the associated Pellian equation, see A134419. The necessary congruence conditions described in A274471 apply here:

(defining x||y to mean x|y and x and y/x are coprime)

if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);

if p^e||n with p=5 or 7 (mod 12), then e is even;

if 3^e||(n+1) with e>0, then e is odd;

if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.

In addition, in order that the Pellian equation has solutions of the correct parity, one must have:

if 2^e||n with e>0, then e is even;

if n is odd, then n=1 (mod 8).

However, these conditions are not sufficient. This sequence consists of the numbers n that satisfy all of the congruence conditions but for which there is no sequence of n consecutive positive odd squares whose sum is a square.

The term 4 is present despite the Pellian equation having a solution with the correct parity, because it leads only to (-1)^2 + 1^2 + 3^2 + 5^2 = 6^2, and the specification of A001033 disallows squares of negative numbers. In every other case the Pellian equation lacks solutions with the right parity. Note however that it may still have solutions with the opposite parity (this can happen only if n=1 mod 8) and so this sequence is not a subsequence of A274471.