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URL: https://oeis.org/A075118

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A075118
Variant on Lucas numbers: a(n) = a(n-1) + 3*a(n-2) with a(0)=2 and a(1)=1.
5
2, 1, 7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, 22258, 51169, 117943, 271450, 625279, 1439629, 3315466, 7634353, 17580751, 40483810, 93226063, 214677493, 494355682, 1138388161, 2621455207, 6036619690, 13900985311, 32010844381, 73713800314, 169746333457
OFFSET
0,1
COMMENTS
The sequence 4,1,7,.. = 2*0^n+A075118(n) is given by trace(A^n) where A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry, Oct 01 2004
For n>2, a(n) is the numerator of the value of the continued fraction 1+3/(1+3/(1+...+3/7)) where there are n-2 1's. - Alexander Mark, Aug 16 2020
REFERENCES
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.
LINKS
Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.
Wikipedia, Lucas sequence
FORMULA
a(n) = ((1+sqrt(13))/2)^n + ((1-sqrt(13))/2)^n.
a(n) = 2*A006130(n) - A006130(n-1) = A075117(3, n).
G.f.: (2-x)/(1-x-3*x^2). - Philippe Deléham, Nov 15 2008
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 13*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = 3^(n/2) * Lucas(n, 1/sqrt(3)). - G. C. Greubel, Jan 15 2020
E.g.f.: 2*exp(x/2)*cosh(sqrt(13)*x/2). - Stefano Spezia, Dec 21 2025
EXAMPLE
a(4) = a(3)+3*a(2) = 10+3*7 = 31.
MAPLE
a:= n-> (Matrix([[1, 2]]). Matrix([[1, 1], [3, 0]])^n)[1, 2]:
seq(a(n), n=0..35); # Alois P. Heinz, Aug 15 2008
MATHEMATICA
a[0]=2; a[1]=1; a[n_]:= a[n]= a[n-1] +3a[n-2]; Table[a[n], {n, 0, 30}]
CoefficientList[Series[(2-x)/(1-x-3x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
LinearRecurrence[{1, 3}, {2, 1}, 40] (* Harvey P. Dale, Jun 18 2017 *)
Table[Round[Sqrt[3]^n*LucasL[n, 1/Sqrt[3]]], {n, 0, 40}] (* G. C. Greubel, Jan 15 2020 *)
PROG
(SageMath) [lucas_number2(n, 1, -3) for n in range(0, 30)] # Zerinvary Lajos, Apr 30 2009
(Magma) I:=[2, 1]; [n le 2 select I[n] else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013
(Magma) R<x>:=PowerSeriesRing(Integers(), 33); Coefficients(R!((2-x)/(1-x-3*x^2))); // Marius A. Burtea, Jan 15 2020
(PARI) my(x='x+O('x^30)); Vec((2-x)/(1-x-3*x^2)) \\ G. C. Greubel, Dec 21 2017
(PARI) polsym(x^2-x-3, 44) \\ Joerg Arndt, Jan 22 2023
(GAP) a:=[2, 1];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Sep 02 2002
STATUS
approved