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A072263
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a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=2, a(1)=3.
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8
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2, 3, 19, 72, 311, 1293, 5434, 22767, 95471, 400248, 1678099, 7035537, 29497106, 123669003, 518492539, 2173822632, 9113930591, 38210904933, 160202367754, 671661627927, 2815996722551, 11806298307288, 49498878534619, 207528127140297
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OFFSET
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0,1
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COMMENTS
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Pisano period lengths: 1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12... - R. J. Mathar, Aug 10 2012
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LINKS
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FORMULA
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a(n) = ((3 + sqrt(29))/2)^n + ((3 - sqrt(29))/2)^n.
G.f.: (2-3*x)/(1-3*x-5*x^2). - R. J. Mathar, Feb 06 2010
Limit_{k -> Infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2.
G.f.: G(0), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 29*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = 5^((n-1)/2)*( 2*sqrt(5)*Fibonacci(n+1, 3/sqrt(5)) - 3*Fibonacci(n, 3/sqrt(5)) ). - G. C. Greubel, Jan 14 2020
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EXAMPLE
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a(5)=5*b(4)+b(6): 1293=5*57+1008.
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MAPLE
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seq(coeff(series((2-3*x)/(1-3*x-5*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 14 2020
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MATHEMATICA
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LinearRecurrence[{3, 5}, {2, 3}, 40] (* Harvey P. Dale, Nov 23 2018 *)
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PROG
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(Sage) [lucas_number2(n, 3, -5) for n in range(0, 16)] # Zerinvary Lajos, Apr 30 2009
(PARI) my(x='x+O('x^40)); Vec((2-3*x)/(1-3*x-5*x^2)) \\ G. C. Greubel, Jan 14 2020
(Magma) I:=[2, 3]; [n le 2 select I[n] else 3*Self(n-1) +5*Self(n-2): n in [1..40]]; // G. C. Greubel, Jan 14 2020
(GAP) a:=[2, 3];; for n in [3..40] do a[n]:=3*a[n-1]+5*a[n-2]; od; a; # G. C. Greubel, Jan 14 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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