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 A230239 Values of N for which the equation x^2 - 4*y^2 = N has integer solutions. 5
 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 (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. 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)). 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=, 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

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)