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%I #37 Aug 09 2024 05:04:06
%S 1,3,63,261,6141,25543,601723,2502921,58962681,245260683,5777740983,
%T 24033043981,566159653621,2354993049423,55477868313843,
%U 230765285799441,5436264935102961,22612643015295763,532698485771776303,2215808250213185301,52199015340698974701
%N Indices of octagonal numbers which are also triangular.
%C From _Ant King_, Nov 01 2011: (Start)
%C lim_{n -> oo} a(2n+1)/a(2n) = (1/5)*(59 + 24*sqrt(6)).
%C lim_{n -> oo} a(2n)/a(2n-1) = (1/5)*(11 + 4*sqrt(6)).
%C (End)
%H Colin Barker, <a href="/A046181/b046181.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalTriangularNumber.html">Octagonal Triangular Number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).
%F For n odd, a(n+2) = 98*a(n+1) - a(n) - 32; for n even, a(n+1) = 49*a(n) - 16 + 10*sqrt(24*a(n)^2 - 16*a(n) + 1). - _Richard Choulet_, Oct 03 2007, Oct 09 2007
%F From _Ant King_, Nov 01 2011: (Start)
%F a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).
%F a(n) = 98*a(n-2) - a(n-4) - 32.
%F a(n) = (1/24)*sqrt(2)((sqrt(6) - (-1)^n)*(sqrt(3) + sqrt(2))^(2*n - 1) + (sqrt(6) + (-1)^n)*(sqrt(3) - sqrt(2))^(2*n - 1) + 4*sqrt(2)).
%F a(n) = ceiling((1/24)*sqrt(2)*(sqrt(6) - (-1)^n)*(sqrt(3) + sqrt(2))^(2*n - 1)).
%F G.f.: x*(1 + 2*x - 38*x^2 + 2*x^3 + x^4)/((1 - x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)).
%F (End)
%t LinearRecurrence[{1, 98, -98, -1, 1}, {1, 3, 63, 261, 6141}, 18] (* _Ant King_, Nov 01 2011 *)
%o (PARI) Vec(-x*(x^4+2*x^3-38*x^2+2*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^50)) \\ _Colin Barker_, Jun 23 2015
%Y Cf. A046182, A046183.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
%E More terms from _Richard Choulet_, Oct 03 2007