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%I #28 Mar 11 2024 23:10:43
%S 1,0,5,5,30,55,205,480,1505,3905,11430,30955,88105,242880,683405,
%T 1897805,5314830,14803855,41378005,115397280,322287305,899273705,
%U 2510710230,7007078755,19560629905,54596023680,152399173205,425379291605,1187375157630,3314271615655
%N a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.
%C This sequence {a(n)} appears in the formula for powers of phi21 := (1 + sqrt(21))/2 = A222134 = 2.791287..., together with b(n) = A015440(n-1), with A015440(-1) = 0, as phi21^n = a(n) + b(n)*phi21(n), for n >= 0. But the later given formulas in terms of scaled Chebyshev polynomials, called here {S21(n)}, are valid also for negative n values, i.e., for nonnegative powers of 1/phi21 = (-1 + sqrt(21))/10 = 0.35825756949... = A367453.
%C Limit_{n->oo} a(n)/a(n-1) = (1 + sqrt(21))/2 = A222134 = 2.791287...
%H Paolo Xausa, <a href="/A365824/b365824.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,5).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.
%F G.f.: (1 - x)/(1 - x - 5*x^2).
%F a(n) = S21(n+1) - S21(n), for n >= 0, where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
%F The above mentioned sequence {b(n)} has terms b(n) = A015440(n-1) = S21(n), for n >= 0, with the same recurrence as {a(n)} but with b(0) = 0 and b(1) = 1, and g.f. x/(1 - x - 5*x^2).
%F The formula for negative indices of S is: S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.
%e phi21^2 = a(2) + b(2)*phi(n) = 5 + phi21 = 7.79128784..., a real algebraic integer in Q(sqrt(21)).
%e (1/phi21)^2 = a(-2) + b(-2)*phi21 = (1/25)*(6 - phi21) = 0.12834848..., a real algebraic number in Q(sqrt(21)).
%t LinearRecurrence[{1,5},{1,0},50] (* _Paolo Xausa_, Nov 21 2023 *)
%o (PARI) a(n) = abs([1, 3; 1, -2]^(n-2)*[5; 5])[2, 1] \\ _Thomas Scheuerle_, Nov 20 2023
%Y Cf. A010477 (sqrt(21)), A015440, A049310, A222134, A367453.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Nov 20 2023