A230240 Values of N for which the equation x^2 - 9*y^2 = N has integer solutions.

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

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.

Also numbers that are congruent to {0,1,4,7,9,13,16,19,25,27,28,31} mod 36. - Keyang Li, Apr 05 2024

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 A352272 A229848

Adjacent sequences: A230237 A230238 A230239 * A230241 A230242 A230243

Keyword

nonn,easy

Author

Colin Barker, Oct 13 2013

Status

approved