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 A134419 Numbers n for which the generalized Pell equation x^2 - n*y^2 = n(n-1)(n+1)/3 has an integer solution for x and y. 12
 1, 2, 4, 11, 16, 23, 24, 25, 26, 33, 47, 49, 50, 52, 59, 64, 73, 74, 88, 96, 97, 100, 107, 121, 122, 146, 148, 169, 177, 184, 191, 193, 194, 196, 218, 239, 241, 242, 244, 249, 256, 276, 289, 292, 297, 299, 311, 312, 313, 337, 338, 347, 352, 361, 362, 376, 383 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS Christopher E. Thompson, Table of n, a(n) for n = 1..13437 (up to 250000, extends first 200 terms computed by T. D. Noe). MATHEMATICA t={}; n=0; While[Length[t]<200, n++; If[Reduce[x^2-n*y^2==n(n^2-1)/3, {x, y}, Integers] =!= False, AppendTo[t, n]]]; t CROSSREFS Cf. A001032, A001033, A274471. Sequence in context: A330856 A180384 A023168 * A342228 A024819 A194545 Adjacent sequences: A134416 A134417 A134418 * A134420 A134421 A134422 KEYWORD nonn AUTHOR T. D. Noe, Oct 25 2007 STATUS approved

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Last modified September 24 19:02 EDT 2023. Contains 365581 sequences. (Running on oeis4.)