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Search: id:A134419
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| A134419 |
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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. |
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+0
2
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| 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
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OFFSET
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1,2
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COMMENT
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..200
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MATHEMATICA
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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
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CROSSREFS
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Sequence in context: A038193 A002382 A023168 this_sequence A072074 A024819 A005822
Adjacent sequences: A134416 A134417 A134418 this_sequence A134420 A134421 A134422
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Oct 25 2007
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