A048922 - OEIS

%I #31 Feb 16 2025 08:32:40

%S 1,425,286209,192904201,130017145025,87631362842409,59063408538638401,

%T 39808649723679439625,26830970850351403668609,

%U 18084034544487122393202601,12188612452013470141614884225

%N Indices of 9-gonal numbers which are also octagonal.

%C 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 Vincenzo Librandi, <a href="/A048922/b048922.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalOctagonalNumber.html">Nonagonal Octagonal Numbers</a>.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (675,-675,1).

%F G.f.: -x*(1 - 250*x + 9*x^2) / ( (x-1)*(x^2 - 674*x + 1) ). - _R. J. Mathar_, Dec 21 2011

%F From _Ant King_, Jan 03 2012: (Start)

%F a(n) = 674*a(n-1) - a(n-2) - 240.

%F a(n) = (1/84)*((sqrt(6) + 3*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3) + (sqrt(6) - 3*sqrt(7))*(sqrt(6) - sqrt(7))^(4*n-3) + 30).

%F a(n) = ceiling((1/84)*(sqrt(6) + 3*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3)). (End)

%t LinearRecurrence[{675,-675,1},{1,425,286209},30] (* _Vincenzo Librandi_, Dec 23 2011 *)

%t Join[{1},Transpose[NestList[{Last[#],674Last[#]-First[#]-240}&, {1,425}, 10]][[2]]] (* _Harvey P. Dale_, Feb 05 2012 *)

%o (Magma) I:=[1, 425, 286209]; [n le 3 select I[n] else 675*Self(n-1)-675*Self(n-2)+1*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Dec 23 2011

%Y Cf. A048923, A048924.

%K nonn,easy,changed

%O 1,2

%A _Eric W. Weisstein_