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%I #24 Apr 26 2021 20:52:56
%S 5,13,41,145,545,2113,8321,33025,131585,525313,2099201,8392705,
%T 33562625,134234113,536903681,2147549185,8590065665,34360000513,
%U 137439477761,549756862465,2199025352705,8796097216513,35184380477441
%N a(n) = 2 * (4^n + 2^n) + 1.
%C 1. Begin with a square tile.
%C 2. Place square tiles on each edge to form a diamond shape.
%C 3. Count the tiles: a(0) = 5.
%C 4. Add tiles to fill the enclosing square.
%C 5. Go to step 2.
%H Iain Fox, <a href="/A085601/b085601.txt">Table of n, a(n) for n = 0..1660</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8)
%F From _R. J. Mathar_, Apr 20 2009: (Start)
%F a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
%F G.f.: -(5 - 22*x + 20*x^2)/((x - 1)*(2*x - 1)*(4*x - 1)).
%F (End)
%F E.g.f.: e^x + 2*(e^(2*x) + e^(4*x)). - _Iain Fox_, Dec 30 2017
%t Table[2(4^n+2^n)+1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{5,13,41},30] (* _Harvey P. Dale_, Dec 30 2017 *)
%o (PARI) first(n) = Vec((5 - 22*x + 20*x^2)/(1 - 7*x + 14*x^2 - 8*x^3) + O(x^n)) \\ _Iain Fox_, Dec 30 2017
%Y Cf. A028403, A046142, A056640, A064412.
%Y Cf. A343175 (essentially the same).
%K nonn,easy
%O 0,1
%A Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jul 07 2003
%E Edited by _Franklin T. Adams-Watters_ and _Don Reble_, Aug 15 2006
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