A015524 a(n) = 3*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.
0, 1, 3, 16, 69, 319, 1440, 6553, 29739, 135088, 613437, 2785927, 12651840, 57457009, 260933907, 1185000784, 5381539701, 24439624591, 110989651680, 504046327177, 2289066543291, 10395523920112, 47210037563373, 214398780130903, 973666603336320, 4421791270925281, 20081040036130083
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,7).
Programs
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Magma
[n le 2 select n-1 else 3*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012
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Mathematica
a[n_]:=(MatrixPower[{{1,3},{1,-4}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *) LinearRecurrence[{3,7},{0,1},30] (* Harvey P. Dale, Jul 04 2011 *) -
PARI
x='x+O('x^30); concat([0], Vec(x/(1 - 3*x - 7*x^2))) \\ G. C. Greubel, Jan 01 2018 -
Sage
[lucas_number1(n,3,-7) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009
Formula
From R. J. Mathar, Apr 21 2008: (Start)
O.g.f.: x/(1 - 3*x - 7*x^2).
a(n) = 14^n*(1/A^n -(-1)^n/B^n)/sqrt(37), where A = sqrt(37) - 3 = A010491 - 3 and B = sqrt(37) + 3 = A010491 + 3. (End)
a(n) = (7*(111+23*sqrt(37))*(1/2*(3+sqrt(37)))^n + (2553 + 431*sqrt(37)) * (1/2 (3-sqrt(37)))^n)/(518*(45+8*sqrt(37))). - Harvey P. Dale, Jul 04 2011
Comments