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%I #13 Sep 08 2022 08:46:10
%S 52,10136,1966380,381467632,74002754276,14356152861960,
%T 2785019652466012,540279456425544416,104811429526903150740,
%U 20332877048762785699192,3944473336030453522492556,765207494312859220577856720,148446309423358658338581711172
%N Numbers n such that the sum of the hexagonal numbers H(n) and H(n+1) is equal to the sum of the pentagonal numbers P(m) and P(m+1) for some m.
%C Also nonnegative integers x in the solutions to 4*x^2-3*y^2+2*x-2*y = 0, the corresponding values of y being A251991.
%H Colin Barker, <a href="/A251990/b251990.txt">Table of n, a(n) for n = 1..437</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F a(n) = 195*a(n-1)-195*a(n-2)+a(n-3).
%F G.f.: 4*x*(x-13) / ((x-1)*(x^2-194*x+1)).
%F a(n) = (-6+(3-2*sqrt(3))*(97+56*sqrt(3))^(-n)+(3+2*sqrt(3))*(97+56*sqrt(3))^n)/24. - _Colin Barker_, Mar 02 2016
%F a(n) = 194*a(n-1)-a(n-2)+48. - _Vincenzo Librandi_, Mar 03 2016
%e 52 is in the sequence because H(52)+H(53) = 5356+5565 = 10921 = 5370+5551 = P(60)+P(61).
%t LinearRecurrence[{195, -195, 1}, {52, 10136, 1966380}, 30] (* _Vincenzo Librandi_, Mar 03 2016 *)
%o (PARI) Vec(4*x*(x-13)/((x-1)*(x^2-194*x+1)) + O(x^100))
%o (Magma) I:=[52,10136]; [n le 2 select I[n] else 194*Self(n-1)- Self(n-2)+48: n in [1..20]]; // _Vincenzo Librandi_, Mar 03 2016
%Y Cf. A000326, A000384, A251991.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Dec 12 2014