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%I #14 Jan 29 2024 05:16:16
%S 1,0,7,7,56,105,497,1232,4711,13335,46312,139657,463841,1441440,
%T 4688327,14778407,47596696,151045545,484222417,1541541232,4931098151,
%U 15721886775,50239573832,160292781257,511969798081,1634019266880,5217807853447,16655942721607,53180597695736,169772196746985
%N Expansion of (1 - x)/(1 - x - 7*x^2).
%C a(n) appears in the formula for powers of the fundamental algebraic number c = (1 + sqrt(29))/2 = A223140 of the quadratic number field Q(sqrt(29)): c^n = a(n) + A015442(n), for n >= 0. The formulas given below and in A015442 in terms of S-Chebyshev polynomials are valid also for c^(-n), for n >= 0, with 1/c = (-1 + sqrt(29))/14 = A367454.
%C a(n) is the number of compositions (ordered partitions) of n into parts >= 2 and there are 7 sorts of each part. - _Joerg Arndt_, Jan 16 2024
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,7).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F a(n) = a(n-1) + 7*a(n-2), with a(0) = 1, a(1) = 0.
%F G.f.: (1 - x)/(1 - x - 7*x^2).
%F a(n) = 7*A015442(n-1), with A015442(-1) = 1/7.
%F a(n) = 7*(-i*sqrt(7))^(n-2)*S(n-2, i/sqrt(7)), with i = sqrt(-1) and the S-Chebyshev polynomial (see A049310). S(-2, x) = -1 and S(-1, x) = 0. The Fibonacci polynomials are F(n, x) = (-i)^(n-1)*S(n-1, i*x).
%t LinearRecurrence[{1,7},{1,0},30] (* _James C. McMahon_, Jan 16 2024 *)
%Y Cf.: A010484, A015442 (partial sums), A049310, A223140, A367454.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Jan 16 2024
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