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Indices of 9-gonal numbers which are also triangular.
5

%I #31 Jul 04 2024 14:17:21

%S 1,10,154,2449,39025,621946,9912106,157971745,2517635809,40124201194,

%T 639469583290,10191389131441,162422756519761,2588572715184730,

%U 41254740686435914,657487278267789889,10478541711598202305,166999180107303446986,2661508340005256949466

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

%C Entries are == 1 (mod 3). - _N. J. A. Sloane_, Sep 22, 2007

%C lim(n -> Infinity, a(n)/a(n-1)) = 8 + 3*sqrt(7). - _Ant King_, Nov 03 2011

%H Colin Barker, <a href="/A048907/b048907.txt">Table of n, a(n) for n = 1..832</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonagonalTriangularNumber.html">Nonagonal Triangular Number.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,-17,1).

%F G.f.: x*(1-7*x+x^2)/((1-x)*(1-16*x+x^2)).

%F a(n+2) = 16*a(n+1)-a(n)-5, a(n+1) = 8*a(n)-2.5+1.5*(28*a(n)^2-20*a(n)+1)^0.5. - _Richard Choulet_, Sep 22 2007

%F From _Ant King_, Nov 03 2011: (Start)

%F a(n) = 17*a(n-1) - 17*a(n-2) + a(n-3).

%F a(n) = ceiling(3/28*(3-sqrt(7))*(8 + 3*sqrt(7))^n).

%F (End)

%F a(n) = A097830(n-1)-7*A097830(n-2)+A097830(n-3). - _R. J. Mathar_, Jul 04 2024

%t LinearRecurrence[{17, -17, 1}, {1, 10, 154}, 17]; (* _Ant King_, Nov 03 2011 *)

%o (PARI) Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-16*x+1)) + O(x^20)) \\ _Colin Barker_, Jun 22 2015

%Y Cf. A001080, A073352, A048908, A048909.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_