A328990 a(n) = (3*b(n) + b(n-1) + 1)/2, where b = A005409.
2, 7, 19, 48, 118, 287, 695, 1680, 4058, 9799, 23659, 57120, 137902, 332927, 803759, 1940448, 4684658, 11309767, 27304195, 65918160, 159140518, 384199199, 927538919, 2239277040, 5406093002, 13051463047, 31509019099, 76069501248, 183648021598, 443365544447
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Bill Allombert, Nicolas Brisebarre, and Alain Lasjaunias, On a two-valued sequence and related continued fractions in power series fields, arXiv:1607.07235 [math.NT], 2016-2017; The Ramanujan Journal 45.3 (2018): 859-871. See Theorem 3.
- Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).
Programs
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Magma
I:=[2,7,19]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-Self(n-3): n in [1..40]] // Vincenzo Librandi, Nov 11 2019
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Maple
m:=35; S:=series( x*(2+x)/((1-x)*(1-2*x-x^2)), x, m+1): seq(coeff(S, x, j), j=1..m); # G. C. Greubel, Apr 23 2021
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Mathematica
LinearRecurrence[{3,-1,-1},{2,7,19},40] (* or *) CoefficientList[Series[(2-x-3x^2-x^3)/(1-x-x^2)/(1-3*x+x^2+x^3),{x,0,33}],x] (* Vincenzo Librandi, Nov 11 2019 *) (LucasL[Range[35] +2, 2] -6)/4 (* G. C. Greubel, Apr 23 2021 *) -
PARI
Vec(x*(2+x)/((1-x)*(1 -2*x -x^2)) + O(x^40)) \\ Colin Barker, Nov 10 2019
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Sage
[(lucas_number2(n+2,2,-1) -6)/4 for n in (1..35)] # G. C. Greubel, Apr 23 2021
Formula
From Colin Barker, Nov 10 2019: (Start)
G.f.: x*(2 + x)/((1 - x)*(1 - 2*x - x^2)).
a(n) = 3*a(n-1) - a(n-2) - a(n-3) for n>3.
a(n) = (-6 + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2))) / 4.
(End)
E.g.f.: (1/2)*exp(x)*(-3 + 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Nov 11 2019
2*a(n) = A001333(n+2) - 3. - R. J. Mathar, Jun 17 2020
a(n) = (A002203(n+2) - 6)/4. - G. C. Greubel, Apr 23 2021