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A084917
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Positive numbers n such that n = 3*y^2 - x^2 with integers x, y.
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10
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2, 3, 8, 11, 12, 18, 23, 26, 27, 32, 39, 44, 47, 48, 50, 59, 66, 71, 72, 74, 75, 83, 92, 98, 99, 104, 107, 108, 111, 122, 128, 131, 138, 143, 146, 147, 156, 162, 167, 176, 179, 183, 188, 191, 192, 194, 200, 207, 218, 219, 227, 234, 236, 239, 242, 243, 251, 263, 264, 275, 282, 284
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OFFSET
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1,1
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COMMENTS
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Positive integers k such that x^2 - 4xy + y^2 + k = 0 has integer solutions.
Comments on method used, from Colin Barker, Jun 06 2014: (Start)
In general, we want to find the values of f, from 1 to 400 say, for which x^2 + bxy + y^2 + f = 0 has integer solutions for a given b.
In order to solve x^2 + bxy + y^2 + f = 0 we can solve the Pellian equation x^2 - Dy^2 = N, where D = (b*b - 4) and N = 4*(b*b - 4)*f.
But since sqrt(D) < N, the classical method of solving x^2 - Dy^2 = N does not work. So I implemented the method described in the 1998 sci.math reference, which says:
"There are several methods for solving the Pellian equation when |N| > sqrt(d). One is to use a brute-force search. If N < 0 then search on y = sqrt(abs(n/d)) to sqrt((abs(n)(x1 + 1))/(2d)) and if N > 0 search on y = 0 to sqrt((n(x1 - 1))/(2d)) where (x1, y1) is the minimum positive solution (x, y) to x^2 - dy^2 = 1. If N < 0, for each positive (x, y) found by the search, also take (-x, y). If N > 0, also take (x, -y). In either case, all positive solutions are generated from these using (x1, y1) in the standard way."
Incidentally all my Pell code is written in B-Prolog, and is somewhat voluminous. (End)
Also, positive integers of the form -x^(+2xy) + 2y^2 of discriminant 12. - N. J. A. Sloane, May 31 2014
The equivalent sequence for x^2 - 3xy + y^2 + k = 0 is A031363.
The equivalent sequence for x^2 - 5xy + y^2 + k = 0 is A237351.
A positive n does not appear in this sequence if and only if there is no integer solution of x^2 - 3*y^2 = -n with i) 0 < y^2 <= n/2 and ii) 0 <= x^2 <= n/2. See the Nagell reference Theorems 108 a and 109, pp. 206-7, with D = 3, N = n and (x_1,y_1) = (2,1). - Wolfdieter Lang, Jan 09 2015
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REFERENCES
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T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
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LINKS
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Table of n, a(n) for n=1..62.
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998 (Edited and cached copy)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
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EXAMPLE
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11 is in the sequence because 3 * 3^2 - 4^2 = 27 - 16 = 11.
12 is in the sequence because 3 * 4^2 - 6^2 = 48 - 36 = 12.
13 is not in the sequence because there is no solution in integers to 3y^2 - x^2 = 13.
From Wolfdieter Lang, Jan 09 2015: (Start)
Referring to the Jan 09 2015 comment above.
n = 1 is out because there is no integer solution of i) 0 < y^2 <= 1/2.
For n = 4, 5, 6, and 7 one has y = 1, x = 0, 1 (and the negative of this). But x^2 - 3 is not -n for these n and x values. Therefore, these n values are missing.
For n = 8 .. 16 one has y = 1, 2 and x = 0, 1, 2. Only y = 2 has a chance and only for n = 8, 11 and 12 the x value 2, 1 and 0, respectively, solves x^2 - 12 = -n. Therefore 9, 10, 13, 14, 15, 16 are missing.
... (End)
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MATHEMATICA
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r[n_] := Reduce[n == 3*y^2 - x^2 && x > 0 && y > 0, {x, y}, Integers]; Reap[For[n = 1, n <= 1000, n++, rn = r[n]; If[rn =!= False, Print["n = ", n, ", ", rn /. C[1] -> 1 // Simplify]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 21 2016 *)
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CROSSREFS
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Cf. A001835 (k = 2), A001075 (k = 3), A237250 (k = 11), A003500 (k = 12), A082841 (k = 18), A077238 (k = 39).
Cf. A031363, A237351.
A141123 gives the primes.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Sequence in context: A005727 A118089 A201541 * A134713 A293910 A173269
Adjacent sequences: A084914 A084915 A084916 * A084918 A084919 A084920
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KEYWORD
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nonn,easy
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AUTHOR
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Roger Cuculière, Jul 14 2003
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EXTENSIONS
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Terms 26 and beyond from Colin Barker, Feb 06 2014
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STATUS
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approved
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