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A217052 a(n) = 3*a(n-1) + 24*a(n-2) + a(n-3), with a(0)=a(1)=1, and a(2)=19.

%I A217052 #20 Feb 19 2025 05:43:54
%S A217052 1,1,19,82,703,4096,29242,186733,1266103,8309143,55500634,367187437,
%T A217052 2441886670,16193659132,107553444913,713750040577,4738726458775,
%U A217052 31453733795086,208804386436435,1386041496850144,9200883498819958,61076450807299765,405436597890428431
%N A217052 a(n) = 3*a(n-1) + 24*a(n-2) + a(n-3), with a(0)=a(1)=1, and a(2)=19.
%C A217052 The Ramanujan type sequence number 10 for the argument 2*Pi/9 defined by the relation a(n) = ((1/3 - c(1))^n + (1/3 - c(2))^n + (1/3 - c(4))^n)*3^(n-1), where c(j) := 2*cos(2*Pi*j/9). We note that c(4) = -cos(Pi/9). The conjugate with a(n) are sequences A217053 and A217069.
%C A217052   For more informations about connections a(n) with these two sequences - see comments in A217053.
%C A217052 The 3-valuation of the sequence a(n) is equal to (1).
%D A217052 Roman Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012, in review.
%H A217052 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,24,1).
%F A217052 G.f.: (1-2*x-8*x^2)/(1-3*x-24*x^2-x^3).
%e A217052 We have a(4)=37*a(2) and a(5) = 2^(12), which implies (1/3 - c(1))^4 + (1/3 - c(2))^4 + (1/3 - c(4))^4 = (37/9)*((1/3 - c(1))^2 + (1/3 - c(2))^2 + (1/3 - c(4))^2) = (37/27)*19 = 703/27, (1/3 - c(1))^5 + (1/3 - c(2))^5 + (1/3 - c(4))^5 = (8/3)^4. Moreover we have a(10) = 676837*a(3).
%t A217052 LinearRecurrence[{3,24,1}, {1,1,19}, 30]
%o A217052 (PARI) Vec((1-2*x-8*x^2)/(1-3*x-24*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%Y A217052 Cf. A217053, A217069, A214699, A214779, A214778, A214951, A214954, A215560, A214569, A215572.
%K A217052 nonn,easy
%O A217052 0,3
%A A217052 _Roman Witula_, Sep 25 2012