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A082975 Denominators of continued fraction convergents to (sqrt(37)-4)/3.

%I A082975 #32 Dec 07 2017 22:20:49
%S A082975 1,1,3,10,13,36,121,157,435,1462,1897,5256,17665,22921,63507,213442,
%T A082975 276949,767340,2578969,3346309,9271587,31161070,40432657,112026384,
%U A082975 376511809,488538193,1353588195,4549302778,5902890973,16355084724
%N A082975 Denominators of continued fraction convergents to (sqrt(37)-4)/3.
%H A082975 Vincenzo Librandi, <a href="/A082975/b082975.txt">Table of n, a(n) for n = 0..200</a>
%H A082975 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 12, 0, 0, 1).
%F A082975 Empirical g.f.: 1 + x*(1+3*x+10*x^2+x^3+x^5)/(1-12*x^3-x^6). - _Colin Barker_, Jan 04 2012
%F A082975 From _Michael Somos_, Dec 07 2017: (Start)
%F A082975 G.f.:  (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6).
%F A082975 Let s := sqrt(37), c := 6 + s, b(0) = 1/2 + 2/s, b(1) = 1/2 + 7/2/s, b(2) := 3/2 + 9/s, then a(3*n + k) ~ b(k) * c^n for k=0, 1, 2.
%F A082975 0 = a(n)*(+3*a(n+1) +5*a(n+2) -3*a(n+3)) +a(n+1)*(+7*a(n+1) +10*a(n+2) -7*a(n+3)) +a(n+2)*(-5*a(n+2) +3*a(n+3)) for all n in Z.
%F A082975 a(n+3) = 12*a(n) + a(n-3),  a(n+1) = a(n-1) + (1+mod(n, 3))*a(n) for all n in Z.
%F A082975 a(3*n - 1) = -(-1)^n * a(-3*n - 1) = 3 * A041061(n-1) for all n in Z. (End)
%t A082975 Denominator[Convergents [(Sqrt[37]-4)/3, 40]] (* _Vincenzo Librandi_, Feb 01 2014 *)
%t A082975 a[ n_] := If[ n < -1, SeriesCoefficient[ (1 - 2*x + 3*x^2 + x^3 + x^4) / (1 + 12*x^3 - x^6), {x, 0, -2 - n}], SeriesCoefficient[ (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6), {x, 0, n}]]; (* _Michael Somos_, Dec 07 2017 *)
%o A082975 (PARI) {a(n) = if( n<-1, n = -2-n; polcoeff( (1 - 2*x + 3*x^2 + x^3 + x^4) / (1 + 12*x^3 - x^6) + x * O(x^n), n), polcoeff( (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6) + x * O(x^n), n))}; /* _Michael Somos_, Dec 07 2017 */
%Y A082975 Numerators are A082962.
%Y A082975 Cf. A041061.
%K A082975 nonn,frac,easy
%O A082975 0,3
%A A082975 _Gary W. Adamson_, May 27 2003
%E A082975 Edited by _Don Reble_, Nov 04 2005
%E A082975 Prepended a(0)=1 from _Vincenzo Librandi_, Feb 01 2014