A230239 Values of N for which the equation x^2 - 4*y^2 = N has integer solutions.

0, 1, 4, 5, 9, 12, 13, 16, 17, 20, 21, 25, 28, 29, 32, 33, 36, 37, 41, 44, 45, 48, 49, 52, 53, 57, 60, 61, 64, 65, 68, 69, 73, 76, 77, 80, 81, 84, 85, 89, 92, 93, 96, 97, 100, 101, 105, 108, 109, 112, 113, 116, 117, 121, 124, 125, 128, 129, 132, 133, 137

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.

This sequence is also numbers that are congruent to {0,1,4,5,9,12,13} mod 16.

Links

Bruno Berselli, Table of n, a(n) for n = 1..1000

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

Formula

G.f.: x^2*(x+1)*(3*x+1)*(x^2-x+1)*(x^2+1) / ((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)).

a(n) ~ 16*n/7. - Stefano Spezia, Apr 08 2024

Example

For N=33, the equation x^2 - 4*y^2 = 33 has solutions (X,Y) = (7,2) and (17,8).

Prog

(PARI)
\\ Values of n for which the equation x^2 - d^2*y^2 = n has integer solutions.
\\ e.g. allpellsq(2, 20) gives [0, 1, 4, 5, 9, 12, 13, 16, 17, 20]
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, A230240.

Sequence in context: A059610 A341783 A319606 * A194154 A297291 A269741

Adjacent sequences: A230236 A230237 A230238 * A230240 A230241 A230242

Keyword

nonn,easy

Author

Colin Barker, Oct 13 2013

Status

approved