A233457 Values of n for which the equation x^2 - 16*y^2 = n has integer solutions.

0, 1, 4, 9, 16, 17, 20, 25, 33, 36, 41, 48, 49, 52, 57, 64, 65, 68, 73, 80, 81, 84, 89, 97, 100, 105, 112, 113, 116, 121, 128, 129, 132, 137, 144, 145, 148, 153, 161, 164, 169, 176, 177, 180, 185, 192, 193, 196, 201, 208, 209, 212, 217, 225, 228, 233, 240

Offset

1,3

Comments

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

Links

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

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

Formula

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

Example

33 is in the sequence because the equation x^2 - 16*y^2 = 33 has solutions (X,Y) = (7,1) and (17,4).

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 4, 9, 16, 17, 20, 25, 33, 36, 41, 48, 49, 52, 57, 64}, 60] (* Harvey P. Dale, Sep 06 2014 *)

Prog

(PARI) concat(0, Vec((7*x^15 +5*x^14 +3*x^13 +x^12 +7*x^11 +5*x^10 +3*x^9 +8*x^8 +5*x^7 +3*x^6 +x^5 +7*x^4 +5*x^3 +3*x^2 +x)/(x^16 -x^15 -x +1) + O(x^100)))

Crossrefs

Cf. A042965, A230239, A230240.

Sequence in context: A369129 A368801 A368799 * A313311 A313312 A313313

Adjacent sequences: A233454 A233455 A233456 * A233458 A233459 A233460

Keyword

nonn,easy

Author

Colin Barker, Mar 18 2014

Status

approved