A154244 a(n) = 6*a(n-1) - 2*a(n-2) for n>1; a(1)=1, a(2)=6.

1, 6, 34, 192, 1084, 6120, 34552, 195072, 1101328, 6217824, 35104288, 198190080, 1118931904, 6317211264, 35665403776, 201358000128, 1136817193216, 6418187159040, 36235488567808, 204576557088768, 1154988365396992

Offset: 1

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A126473.

lim_{n -> infinity} a(n)/a(n-1) = 3+sqrt(7) = 5.6457513110....

a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..500
• Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).
• Index entries for linear recurrences with constant coefficients, signature (6,-2).

Crossrefs

Equals 1 followed by 2*A010913 (Pisot sequence E(3,17)).

Cf. A010465 (decimal expansion of square root of 7), A126473.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((3+r)^n-(3-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

Magma

I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 02 2012

Mathematica

a[n_]:=(MatrixPower[{{1,3},{1,5}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{6, -2}, {1, 6}, 40] (* Vincenzo Librandi, Feb 02 2012 *)

Maxima

a[1]:1$ a[2]:6$ a[n]:=6*a[n-1]-2*a[n-2]$ makelist(a[n], n, 1, 21); /* Bruno Berselli, May 30 2011 */

PARI

Vec(1/(1-6*x+2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 28 2011

Sage

[lucas_number1(n,6,2) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = ((3 + sqrt(7))^n - (3 - sqrt(7))^n)/(2*sqrt(7)).

G.f.: x/(1-6*x+2*x^2). - Philippe Deléham, Jan 06 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

Name (corrected) from Philippe Deléham, Jan 06 2009