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%I #22 Sep 08 2022 08:44:58
%S 1,631125,286703855361,130242107189808901,59165603001256545014625,
%T 26877395137662573622784125461,12209701798707362366915379264832801,
%U 5546550074879110936730454426529871893125
%N 9-gonal octagonal numbers.
%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))^8 = 227137 + 35048*sqrt(42). - _Ant King_, Jan 03 2012
%H Vincenzo Librandi, <a href="/A048924/b048924.txt">Table of n, a(n) for n = 1..100</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonagonalOctagonalNumber.html">Nonagonal Octagonal Number</a>.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (454275,-454275,1).
%F a(n) = 454275*a(n-1) - 454275*a(n-2) + a(n-3). - _Vincenzo Librandi_, Dec 24 2011
%F From _Ant King_, Jan 03 2012: (Start)
%F G.f.: x*(1 + 176850*x + 261*x^2) / ((1-x)*(1 - 454274*x + x^2)).
%F a(n) = 454274*a(n-1) - a(n-2) + 177112.
%F a(n) = (1/672)*((11*sqrt(7) - 9*sqrt(6))*(sqrt(6) + sqrt(7))^(8*n-5) - (11*sqrt(7) + 9*sqrt(6))*(sqrt(6) - sqrt(7))^(8*n-5) - 262).
%F a(n) = floor((1/672)*(11*sqrt(7) - 9*sqrt(6))*(sqrt(6) + sqrt(7))^(8*n-5)). (End)
%t LinearRecurrence[{454275,-454275,1},{1,631125,286703855361},30] (* _Vincenzo Librandi_, Dec 24 2011 *)
%o (Magma) I:=[1, 631125, 286703855361]; [n le 3 select I[n] else 454275*Self(n-1)-454275*Self(n-2)+Self(n-3): n in [1..10]]; // _Vincenzo Librandi_, Dec 24 2011
%Y Cf. A048922, A048923.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
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