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A048923 Indices of octagonal numbers which are also 9-gonal.

%I A048923 #25 Feb 16 2025 08:32:40
%S A048923 1,459,309141,208360351,140434567209,94652689938291,63795772583840701,
%T A048923 42998256068818693959,28980760794611215887441,
%U A048923 19532989777311890689441051,13165206129147419713467380709
%N A048923 Indices of octagonal numbers which are also 9-gonal.
%C A048923 As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (sqrt(6) + sqrt(7))^4 = 337 + 52*sqrt(42). - _Ant King_, Jan 03 2012
%H A048923 Vincenzo Librandi, <a href="/A048923/b048923.txt">Table of n, a(n) for n = 1..200</a>
%H A048923 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalOctagonalNumber.html">Nonagonal Octagonal Number.</a>
%H A048923 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (675,-675,1).
%F A048923 G.f.: x*(-1 + 216*x + 9*x^2) / ( (x-1)*(x^2 - 674*x + 1) ). - _R. J. Mathar_, Dec 21 2011
%F A048923 From _Ant King_, Jan 03 2012: (Start)
%F A048923 a(n) = 674*a(n-1) - a(n-2) - 224.
%F A048923 a(n) = (1/168)*((7*sqrt(6) + 2*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3) + (7*sqrt(6) - 2*sqrt(7))*(sqrt(6) - sqrt(7))^(4*n-3) + 56).
%F A048923 a(n) = ceiling((1/168)*(7*sqrt(6) + 2*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3)). (End)
%t A048923 LinearRecurrence[{675,-675,1},{1,459,309141},30] (* _Vincenzo Librandi_, Dec 24 2011 *)
%o A048923 (Magma) I:=[1, 459, 309141]; [n le 3 select I[n] else 675*Self(n-1)-675*Self(n-2)+Self(n-3): n in [1..15]]; // _Vincenzo Librandi_, Dec 24 2011
%Y A048923 Cf. A048922, A048924.
%K A048923 nonn,easy
%O A048923 1,2
%A A048923 _Eric W. Weisstein_