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%I #22 Aug 16 2015 12:03:56
%S 1,2,18,49,529,1458,15842,43681,474721,1308962,14225778,39225169,
%T 426298609,1175446098,12774732482,35224157761,382815675841,
%U 1055549286722,11471695542738,31631254443889,343768050606289,947882084029938,10301569822645922,28404831266454241
%N Indices of 9-gonal numbers that are also square.
%C From _Ant King_, Nov 18 2011: (Start)
%C lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)) = 1/25 * (9 + 2 * sqrt(14))^2.
%C lim( n -> Infinity, a(2n)/a(2n-1)) = 1/25 * (39 + 8 * sqrt(14)).
%C (14 * a(n) - 5)^2 - 56 * A048911(n) ^ 2 = 25.
%C (End)
%H Colin Barker, <a href="/A048910/b048910.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonagonalSquareNumber.html">Nonagonal Square Number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,30,-30,-1,1).
%F From _Ant King_, Nov 18 2011: (Start)
%F a(n) = 30 * a(n - 2) - a(n-4) - 10.
%F a(n) = a(n - 1) + 30 * a(n - 2) - 30 * a(n - 3) - a(n - 4) + a(n - 5).
%F Let p = 9 + 4 * sqrt(2) + sqrt(7) + 2 * sqrt(14) and q = 9 - 4 * sqrt(2) - sqrt(7) + 2 * sqrt(14). Then
%F a(n) = 1/56 * ( ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) + ( p + q * (-1)^n) * ( 2 * sqrt(2) - sqrt(7))^n + 20 ).
%F a(n) = ceiling (1/56 * ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) ).
%F G.f.: x * (1 + x - 14 * x^2 + x^3 + x^4) / ((1 - x) * (1 - 30 * x^2 + x^4)).
%F (End)
%t LinearRecurrence[ {1, 30, - 30, -1, 1 }, {1, 2, 18, 49, 529}, 21 ] (* _Ant King_, Nov 18 2011 *)
%o (PARI) Vec(-x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^4-30*x^2+1)) + O(x^50)) \\ _Colin Barker_, Jun 22 2015
%Y Cf. A048911, A036411.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
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