The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Wikipedia, Lucas sequence Index entries for linear recurrences with constant coefficients, signature (1,3). 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 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 (Sage) [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:=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 Cf. A000032, A006130, A014551, A072265, A075117, A274977. Sequence in context: A032135 A032039 A203991 * A100245 A275320 A272931 Adjacent sequences: A075115 A075116 A075117 * A075119 A075120 A075121 KEYWORD nonn,easy AUTHOR Henry Bottomley, Sep 02 2002 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 13:14 EST 2023. Contains 367591 sequences. (Running on oeis4.)