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Positive integers n such that (n+39)^3 - n^3 is a square.
8

%I #7 Mar 25 2020 16:49:44

%S 26,871,59930,1155895,77814386,1500376111,101003038370,1947487061455,

%T 131101866015146,2527836705417751,170170121084646410,

%U 3281130096145204615,220880686066005050306,4258904336959770197791,286702960343553470676050,5528054548243685571553375

%N Positive integers n such that (n+39)^3 - n^3 is a square.

%H Colin Barker, <a href="/A263945/b263945.txt">Table of n, a(n) for n = 1..642</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1298,-1298,-1,1).

%F a(n) = a(n-1)+1298*a(n-2)-1298*a(n-3)-a(n-4)+a(n-5) for n>5.

%F G.f.: 13*x*(5*x^4+65*x^3-1947*x^2-65*x-2) / ((x-1)*(x^2-36*x-1)*(x^2+36*x-1)).

%e 26 is in the sequence because (26+39)^3 - 26^3 = 507^2.

%t LinearRecurrence[{1,1298,-1298,-1,1},{26,871,59930,1155895,77814386},20] (* _Harvey P. Dale_, Mar 25 2020 *)

%o (PARI) Vec(13*x*(5*x^4+65*x^3-1947*x^2-65*x-2)/((x-1)*(x^2-36*x-1)*(x^2+36*x-1)) + O(x^30))

%Y Cf. A263942 (4), A263943 (21), A263944 (28), A263946 (52), A263947 (57), A263948 (61), A263949 (84) where the parenthesized number is k in the expression (n+k)^3 - n^3.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Oct 30 2015