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A145718 Numbers x such that there exists n in N with (x+127)^3-x^3=n^2. 2

%I

%S 762,1676527,3403477826,6909058439031,14025385227883882,

%T 28471525103545970207,57797181934813091765106,

%U 117328250856145472737323751,238176291440793374843675578202,483497754296559694787188686555087,981500203045724739624618190031377186

%N Numbers x such that there exists n in N with (x+127)^3-x^3=n^2.

%H Vincenzo Librandi, <a href="/A145718/b145718.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2031,-2031,1).

%F a(n+2) = 2030*a(n+1)-a(n)+128778.

%F a(n) = -(127/2)+(1651/4)*{[1015+52*sqrt(381)]^n+[1015-52*sqrt(381)]^n}+(127/6)*sqrt(381)*{[1015+52*sqrt(381)]^n-[1015-52*sqrt(381)]^n} with n>=0. -_Paolo P. Lava_, Nov 25 2008

%F a(n) = 127*A145720(n). - _Colin Barker_, Oct 18 2014

%F G.f.: 127*x*(7*x^2-1015*x-6) / ((x-1)*(x^2-2030*x+1)). - _Colin Barker_, Oct 18 2014

%e a(1)=762 because the first relation is (762+127)^3-762^3=16129^2.

%t CoefficientList[Series[127 (7 x^2 - 1015 x - 6)/((x - 1) (x^2 - 2030 x + 1)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Oct 18 2014 *)

%o (PARI) Vec(127*x*(7*x^2-1015*x-6)/((x-1)*(x^2-2030*x+1)) + O(x^20)) \\ _Colin Barker_, Oct 18 2014

%o (MAGMA) I:=[762,1676527]; [n le 2 select I[n] else 2030*Self(n-1)-Self(n-2)+128778: n in [1..20]]; // _Vincenzo Librandi_, Oct 18 2014

%Y Cf. A145720.

%K easy,nonn

%O 1,1

%A _Richard Choulet_, Oct 16 2008

%E Editing and more terms from _Colin Barker_, Oct 18 2014

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Last modified January 21 13:55 EST 2020. Contains 331113 sequences. (Running on oeis4.)