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%I #17 Jun 22 2015 11:44:46
%S 1,3,33,91,989,2727,29637,81719,888121,2448843,26613993,73383571,
%T 797531669,2199058287,23899336077,65898365039,716182550641,
%U 1974751892883,21461577183153,59176658421451,643131132943949,1773325000750647,19272472411135317,53140573364097959
%N Indices of square numbers which are also 9-gonal.
%C From _Ant King_, Nov 18 2011: (Start)
%C lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)).
%C lim( n -> Infinity, a(2n)/a(2n-1)) = 1/25 * (39 + 8 * sqrt(14)).
%C (End)
%H Colin Barker, <a href="/A048911/b048911.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_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,30,0,-1)
%F From _Ant King_, Nov 18 2011: (Start)
%F a(n) = 30 * a(n-2) - a(n-4).
%F G.f.: x * (1 + x) ^ 3 / (1 - 30 * x ^ 2 + x ^ 4).
%F Let p = 8 * sqrt(7) + 9 * sqrt(14) - 7 * sqrt(2) - 28 and q = 7 * sqrt(2) + 9 * sqrt(14) - 8 * sqrt(7) - 28. Then
%F a(n) = 1/112 * ( ( p + q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7)) ^ n - ( p - q * (-1) ^ n) * ( 2 * sqrt(2) - sqrt(7)) ^ ( n - 1) ).
%F a(n) = floor ( 1/112 * ( p + q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7)) ^ n ).
%F (End)
%t LinearRecurrence[ {0, 30, 0, - 1 }, { 1, 3, 33, 91 } , 21 ] (* _Ant King_, Nov 18 2011 *)
%o (PARI) Vec(x*(x+1)^3/(x^4-30*x^2+1) + O(x^50)) \\ _Colin Barker_, Jun 22 2015
%Y Cf. A048910, A036411.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
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