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%I #36 Mar 08 2023 11:49:05
%S 1,3,4,15,19,72,91,345,436,1653,2089,7920,10009,37947,47956,181815,
%T 229771,871128,1100899,4173825,5274724,19997997,25272721,95816160,
%U 121088881,459082803,580171684,2199597855,2779769539,10538906472
%N Numerators in continued fraction [0; 1, 3, 1, 3, 1, 3, ...].
%C A136210(n)/A136211(n) tends to 0.7912878474... = (sqrt(21) - 3)/2 = continued fraction [0; 1, 3, 1, 3, 1, 3, ...] = the inradius of a right triangle with hypotenuse 5, legs 2 and sqrt(21).
%C This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - _Peter Bala_, May 14 2014
%H J. L. Ramirez, F. Sirvent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Ramirez/ramirez9.html">A q-Analogue of the Bi-Periodic Fibonacci Sequence</a>, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=3, b=1.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-1).
%F a(0) = 0, a(1) = 1, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix [1, 3; 1, 4] = T, [a(2n-1), a(2n)] = top row of T^n.
%F g.f.: x*(1+3*x-x^2)/(1-5*x^2+x^4). - _Colin Barker_, Jan 04 2012
%F a(-n) = -(-1)^n * a(n). a(2*n - 1) = A004253(n). a(2*n) = 3 * A004254(n). - _Michael Somos_, May 15 2014
%F a(n+1) - a(n-1) = a(n) * (2 - (-1)^n) for all n in Z. - _Michael Somos_, May 15 2014
%e a(4) = 15 = 3*a(3) + a(2) = 3*4 + 3.
%e a(5) = 19 = a(4) + a(3) = 15 + 4.
%e T^3 = [19, 72; 24, 91], where [19, 72] = [a(5), a(6)]. [24, 91] = [A136211(5), A136211(6)].
%e G.f. = x + 3*x^2 + 4*x^3 + 15*x^4 + 19*x^5 + 72*x^6 + 91*x^7 + 345*x^8 + ...
%t a = {1, 3}; Do[If[EvenQ[n], AppendTo[a, 3*a[[ -1]] + a[[ -2]]], AppendTo[a, a[[ -1]] + a[[ -2]]]], {n, 3, 30}]; a (* _Stefan Steinerberger_, Dec 31 2007 *)
%t a[n_] := FromContinuedFraction[ Join[{0}, 3 - 2*Array[Mod[#, 2]&, n]]] // Numerator; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, May 15 2014 *)
%o (PARI) {a(n) = (-1)^((n+1) * (n<0)) * polcoeff( x * (1 + 3*x - x^2) / (1 - 5*x^2 + x^4) + x * O(x^abs(n)), abs(n))}; /* _Michael Somos_, May 15 2014 */
%Y Cf. A004253, A004254, A136211.
%K nonn,frac,easy
%O 1,2
%A _Gary W. Adamson_, Dec 21 2007
%E More terms from _Stefan Steinerberger_, Dec 31 2007
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