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A049664 a(n) = (F(6*n+3) - 2)/32, where F = A000045 (the Fibonacci sequence).

%I A049664 #37 Nov 28 2024 14:17:16
%S A049664 0,1,19,342,6138,110143,1976437,35465724,636406596,11419853005,
%T A049664 204920947495,3677157201906,65983908686814,1184033199160747,
%U A049664 21246613676206633,381255012972558648,6841343619829849032,122762930143964723929,2202891398971535181691
%N A049664 a(n) = (F(6*n+3) - 2)/32, where F = A000045 (the Fibonacci sequence).
%C A049664 Partial sums of Chebyshev polynomials S(n,18).
%H A049664 G. C. Greubel, <a href="/A049664/b049664.txt">Table of n, a(n) for n = 0..750</a>
%H A049664 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A049664 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (19,-19,1).
%F A049664 G.f.: x/(1-19*x+19*x^2-x^3) = x/((1-x)*(1-18*x+x^2)).
%F A049664 a(n+1) = Sum_{k=0..n} S(k, 18), with n>=0, S(k, 18) = U(k, 9) = A049660(k+1).
%F A049664 a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=19.
%F A049664 a(n) = 18*a(n-1) - a(n-2) + 1, n>=2, a(0)=0, a(1)=1.
%F A049664 a(n+1) = (S(n+1, 18) - S(n, 18) - 1)/16, n>=0.
%F A049664 a(n) = (1/8)*Sum_{k=0..n} Fibonacci(6*k). - _Gary Detlefs_, Dec 07 2010
%F A049664 Product_{n>=2} (1 - 1/a(n)) = phi^6/19 = (4*sqrt(5)+9)/19, where phi is the golden ratio (A001622). - _Amiram Eldar_, Nov 28 2024
%t A049664 LinearRecurrence[{19, -19, 1}, {0, 1, 19}, 50] (* or *) Table[(Fibonacci[ 6*n +3] - 2)/32, {n,0,30}] (* _G. C. Greubel_, Dec 02 2017 *)
%o A049664 (PARI) a(n)=fibonacci(6*n+3)\32 \\ _Charles R Greathouse IV_, Oct 07 2016
%o A049664 (Magma) [(Fibonacc9(6*n+3)-2)/32: n in [0..30]]; // _G. C. Greubel_, Dec 02 2017
%Y A049664 Cf. A000045, A001622, A049660, A053606.
%Y A049664 Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).
%K A049664 nonn,easy
%O A049664 0,3
%A A049664 _Clark Kimberling_
%E A049664 Chebyshev comments from _Wolfdieter Lang_, Aug 31 2004