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A248205 Indices of centered octagonal numbers (A016754) that are also pentagonal numbers (A000326).

%I A248205 #23 Sep 08 2022 08:46:10
%S A248205 1,50,4851,475300,46574501,4563825750,447208348951,43821854371400,
%T A248205 4294094520048201,420777441110352250,41231895134294472251,
%U A248205 4040304945719747928300,395908652785401002501101,38795007668023578497179550,3801514842813525291721094751
%N A248205 Indices of centered octagonal numbers (A016754) that are also pentagonal numbers (A000326).
%C A248205 Positive integers y in the solutions to 3*x^2 - 8*y^2 - x + 8*y - 2 = 0, the corresponding values of x being A046172.
%H A248205 Colin Barker, <a href="/A248205/b248205.txt">Table of n, a(n) for n = 1..503</a>
%H A248205 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).
%F A248205 a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3).
%F A248205 G.f.: x*(49*x-1) / ((x-1)*(x^2 - 98*x + 1)).
%F A248205 a(n) = (1/2+1/48*(49+20*sqrt(6))^(-n)*(-12-5*sqrt(6)+(-12+5*sqrt(6))*(49+20*sqrt(6))^(2*n))). - _Colin Barker_, Mar 03 2016
%e A248205 50 is in the sequence because the 50th centered octagonal number is 9801, which is also the 81st pentagonal number.
%t A248205 LinearRecurrence[{99, -99, 1}, {1, 50, 4851}, 20] (* _Vincenzo Librandi_, Jun 13 2015 *)
%o A248205 (PARI) Vec(x*(49*x-1)/((x-1)*(x^2-98*x+1)) + O(x^100))
%o A248205 (Magma) I:=[1,50,4851]; [n le 3 select I[n] else 99*Self(n-1)-99*Self(n-2)+Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jun 13 2015
%Y A248205 Cf. A000290, A000326, A046172, A036353.
%K A248205 nonn,easy
%O A248205 1,2
%A A248205 _Colin Barker_, Jan 11 2015