A082975 Denominators of continued fraction convergents to (sqrt(37)-4)/3.
1, 1, 3, 10, 13, 36, 121, 157, 435, 1462, 1897, 5256, 17665, 22921, 63507, 213442, 276949, 767340, 2578969, 3346309, 9271587, 31161070, 40432657, 112026384, 376511809, 488538193, 1353588195, 4549302778, 5902890973, 16355084724
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (0, 0, 12, 0, 0, 1).
Programs
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Mathematica
Denominator[Convergents [(Sqrt[37]-4)/3, 40]] (* Vincenzo Librandi, Feb 01 2014 *) 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 *)
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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 */
Formula
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
From Michael Somos, Dec 07 2017: (Start)
G.f.: (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6).
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.
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.
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.
a(3*n - 1) = -(-1)^n * a(-3*n - 1) = 3 * A041061(n-1) for all n in Z. (End)
Extensions
Edited by Don Reble, Nov 04 2005
Prepended a(0)=1 from Vincenzo Librandi, Feb 01 2014