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A233457 Values of n for which the equation x^2 - 16*y^2 = n has integer solutions. 1
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 (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.

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: A080819 A313309 A313310 * A313311 A313312 A313313

Adjacent sequences:  A233454 A233455 A233456 * A233458 A233459 A233460

KEYWORD

nonn,easy

AUTHOR

Colin Barker, Mar 18 2014

STATUS

approved

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Last modified April 18 14:42 EDT 2019. Contains 322209 sequences. (Running on oeis4.)