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A137221 a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4), with a(0)=0, a(1)=0, a(2)=0, a(3)=1.

%I A137221 #29 Jan 02 2024 08:58:18
%S A137221 0,0,0,1,5,16,43,107,256,597,1365,3072,6827,15019,32768,70997,152917,
%T A137221 327680,699051,1485483,3145728,6640981,13981013,29360128,61516459,
%U A137221 128625323,268435456,559240533,1163220309,2415919104,5010795179
%N A137221 a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4), with a(0)=0, a(1)=0, a(2)=0, a(3)=1.
%H A137221 G. C. Greubel, <a href="/A137221/b137221.txt">Table of n, a(n) for n = 0..1000</a>
%H A137221 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,8,-4).
%F A137221 Binomial transform of A002264; a(n+1) - 2*a(n) = A024495.
%F A137221 From _R. J. Mathar_, Mar 17 2008: (Start)
%F A137221 O.g.f.: x^3/((1-x+x^2)(1-2*x)^2).
%F A137221 a(n) = ( -3*2^n + A001787(n+1) + 2*A010892(n) )/6. (End)
%F A137221 a(n) = (1/3)*(2^(n-1)*(n-2) + ChebyshevU(n, 1/2)). - _G. C. Greubel_, Jan 05 2022
%t A137221 Table[(1/3)*(2^(n-1)*(n-2) + ChebyshevU[n, 1/2]), {n, 0, 40}] (* _G. C. Greubel_, Jan 05 2022 *)
%t A137221 LinearRecurrence[{5,-9,8,-4},{0,0,0,1},40] (* _Harvey P. Dale_, Apr 30 2023 *)
%o A137221 (Magma) [n le 4 select Floor((n-1)/3) else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4): n in [1..30]]; // _G. C. Greubel_, Jan 05 2022
%o A137221 (Sage) [(1/3)*(2^(n-1)*(n-2) + chebyshev_U(n, 1/2)) for n in (0..40)] # _G. C. Greubel_, Jan 05 2022
%Y A137221 Same recurrence as in A100335 (essentially first differences of this sequence).
%Y A137221 Cf. A001787, A002264, A010892, A024495.
%K A137221 nonn
%O A137221 0,5
%A A137221 _Paul Curtz_, Mar 07 2008
%E A137221 More terms from _R. J. Mathar_, Mar 17 2008