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A230240 Values of N for which the equation x^2 - 9*y^2 = N has integer solutions. 6
0, 1, 4, 7, 9, 13, 16, 19, 25, 27, 28, 31, 36, 37, 40, 43, 45, 49, 52, 55, 61, 63, 64, 67, 72, 73, 76, 79, 81, 85, 88, 91, 97, 99, 100, 103, 108, 109, 112, 115, 117, 121, 124, 127, 133, 135, 136, 139, 144, 145, 148, 151, 153, 157, 160, 163, 169, 171, 172 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This equation is a Pellian equation of the form x^2 - D^2*y^2 = N. A042965 covers the case D=1.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

FORMULA

G.f.: x^2*(5*x^11 +3*x^10 +x^9 +2*x^8 +6*x^7 +3*x^6 +3*x^5 +4*x^4 +2*x^3 +3*x^2 +3*x +1) / ((x -1)^2*(x +1)*(x^2 -x +1)*(x^2 +1)*(x^2 +x +1)*(x^4 -x^2 +1)).

EXAMPLE

For N=55, the equation x^2 - 9*y^2 = 55 has solutions (X,Y) = (8,1) and (28,9).

PROG

(PARI)

\\ Values of n for which the equation x^2 - d^2*y^2 = n has integer solutions.

\\ e.g. allpellsq(3, 20) gives [0, 1, 4, 7, 9, 13, 16, 19]

allpellsq(d, nmax) = {

  local(v=[0], n, w);

  for(n=1, nmax,

    w=pellsq(d, n);

    if(#w>0, v=concat(v, n))

  );

  v

}

\\ All integer solutions to x^2-d^2*y^2=n.

\\ e.g. pellsq(5, 5200) gives [265, 51; 140, 24; 85, 9]

pellsq(d, n) = {

  local(m=Mat(), f, x, y);

  fordiv(n, f,

    if(f*f>n, break);

    if((n-f^2)%(2*f*d)==0,

      y=(n-f^2)\(2*f*d);

      x=d*y+f;

      m=concat(m, [x, y]~)

    )

  );

  m~

}

CROSSREFS

Cf. A042965, A230239.

Sequence in context: A310959 A310960 A108287 * A243175 A229848 A239993

Adjacent sequences:  A230237 A230238 A230239 * A230241 A230242 A230243

KEYWORD

nonn

AUTHOR

Colin Barker, Oct 13 2013

STATUS

approved

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Last modified September 19 10:57 EDT 2019. Contains 327192 sequences. (Running on oeis4.)