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A370175 a(n) = floor(x*a(n-1)) for n > 0 where x = (5+3*sqrt(5))/2, a(0) = 1.

%I A370175 #32 May 25 2024 15:28:35
%S A370175 1,5,29,169,989,5789,33889,198389,1161389,6798889,39801389,233001389,
%T A370175 1364013889,7985076389,46745451389,273652638889,1601990451389,
%U A370175 9378215451389,54901029513889,321396224826389,1881486271701389,11014412482638889,64479493771701389
%N A370175 a(n) = floor(x*a(n-1)) for n > 0 where x = (5+3*sqrt(5))/2, a(0) = 1.
%C A370175 x = A090550 = 1 + 3*phi = 5.854101966..., where phi is the golden ratio.
%H A370175 Paolo Xausa, <a href="/A370175/b370175.txt">Table of n, a(n) for n = 0..1000</a>
%H A370175 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,0,-5).
%F A370175 a(n) = 6*a(n-1) - 5*a(n-3), a(0) = 1, a(1) = 5, a(2) = 29.
%F A370175 a(n) = 5*a(n-1) + 5*a(n-2) - 1.
%F A370175 a(n) = (4*(5-2*sqrt(5))*((5-3*sqrt(5))/2)^n + 4*(5+2*sqrt(5))*((5+3*sqrt(5))/2)^n + 5)/45.
%F A370175 G.f.: (1 - x - x^2)/(1 - 6*x + 5*x^3).
%F A370175 a(n) = Sum_{k = 0..n} A370174(n,k)*4^k.
%F A370175 a(n) = (8*A057088(n) + 4*A057088(n-1) + 1)/9.
%e A370175 a(0) = 1, a(1) = floor(x) = 5 where x = (5+3*sqrt(5))/2.
%e A370175 a(2) = floor(5*x) = 29, a(3) = floor(29*x) = 169.
%t A370175 NestList[Floor[#*(5 + 3*Sqrt[5])/2] &, 1, 30] (* or *)
%t A370175 LinearRecurrence[{6, 0, -5}, {1, 5, 29}, 30] (* _Paolo Xausa_, May 25 2024 *)
%Y A370175 Cf. A057088, A090550, A370174.
%K A370175 nonn,easy
%O A370175 0,2
%A A370175 _Philippe Deléham_, Mar 18 2024