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y-values in the solutions to x^2 + x = 5*y^2 + y.
2

%I #35 Sep 08 2022 08:46:12

%S 0,1,52,357,16776,114985,5401852,37024845,1739379600,11921885137,

%T 560074829380,3838809989301,180342355680792,1236084894669817,

%U 58069678454385676,398015497273691805,18698256119956506912,128159754037234091425,6020780400947540840020

%N y-values in the solutions to x^2 + x = 5*y^2 + y.

%C Also, numbers k such that 2*k^2 + k*(k+1)/2 is a triangular number. Example: 114985 is a term because 2*114985^2 + 114985*114986/2 = 257114*257115/2. - _Bruno Berselli_, Mar 02 2018

%H Colin Barker, <a href="/A257940/b257940.txt">Table of n, a(n) for n = 1..798</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,322,-322,-1,1).

%F a(1) = 0, a(2) = 1, a(3) = 52, a(4) = 357, a(5) = 16776; for n > 5, a(n) = a(n-1) + 322*a(n-2) - 322*a(n-3) - a(n-4) + a(n-5).

%F a(n) = 322*a(n-2) - a(n-4) + 32.

%F a(n) = 72*A257939(n-2) + 161*a(n-2) + 52.

%F G.f.: x^2*(3*x^3+17*x^2-51*x-1) / ((x-1)*(x^2-18*x+1)*(x^2+18*x+1)). - _Colin Barker_, May 14 2015

%t LinearRecurrence[{1, 322, -322, -1, 1}, {0, 1, 52, 357, 16776}, 30] (* _Vincenzo Librandi_, May 15 2015 *)

%o (Magma) I:=[0, 1, 52, 357, 16776]; [n le 5 select I[n] else Self(n-1)+322*Self(n-2)-322*Self(n-3)-Self(n-4)+Self(n-5): n in [1..19]];

%o (PARI) concat(0, Vec((3*x^3+17*x^2-51*x-1)/((x-1)*(x^2-18*x+1)*(x^2+18*x+1)) + O(x^100))) \\ _Colin Barker_, May 14 2015

%Y Cf. A257939.

%K nonn,easy

%O 1,3

%A _Arkadiusz Wesolowski_, May 13 2015