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A002605 a(n)=2*(a(n-1)+a(n-2)), a(0)=0, a(1)=1. 103
0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, 154947584, 423324672, 1156544512, 3159738368, 8632565760, 23584608256, 64434348032, 176037912576, 480944521216, 1313964867584 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Individually, both this sequence and A028859 are convergents to 1+sqrt(3). Mutually, both sequences are convergents to 2+sqrt(3) and 1+sqrt(3)/2.- Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001

The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n+1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(4). - Cino Hilliard, Sep 25 2005

The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007

[1,3; 1,1]^n *[1,0] = [A026150(n), a(n)]. - Gary W. Adamson, Mar 21 2008

(1+sqrt(3))^n = A026150(n) + a(n)*sqrt(3) - Gary W. Adamson, Mar 21 2008

a(n+1) is the number of ways to tile a board of length n using red and blue tiles of length one and two. [From Geoffrey Critzer, Feb 07 2009]

Starting with offset 1 = INVERT transform of the Jacobsthal sequence, A001045: (1, 1, 3, 5, 11, 21,...). [From Gary W. Adamson, May 12 2009]

Starting with "1" = INVERTi transform of A007482: (1, 3, 11, 39, 139,...). [From Gary W. Adamson, Aug 06 2010]

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A026150, without the first leading 1. [Johannes W. Meijer, Aug 15 2010]

The sequence 0, 1, -2, 6, -16, 44, -120, 328, -896,.. (with alternating signs) is the Lucas U(-2,-2)-sequence. - R. J. Mathar, Jan 08 2013

REFERENCES

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

A. F. Horadam, Special properties of the sequence w_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), lhs, m=2.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Dale Gerdemann Bird Flock

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 476

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (2,2).

Index entries for sequences related to Chebyshev polynomials.

Index entries for Lucas sequences.

FORMULA

Wolfdieter Lang observes that a(n)=(-I*sqrt(2))^n*U(n, I/sqrt(2)) where U(n, x) is the Chebyshev U-polynomial.

G.f.: x/(1-2*x-2*x^2).

E.g.f. exp(x)(sinh(sqrt(3)x)/sqrt(3)+cosh(sqrt(3)x)); a(n)=(1+sqrt(3))^n(1/2+sqrt(3)/6)+(1-sqrt(3))^n(1/2-sqrt(3)/6). Binomial transform of 1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. - Paul Barry, Sep 17 2003

a(n)=(1/3)*Sum(k, 1, 5, Sin(Pi*k/2)Sin(Pi*k/3)(1+2Cos(Pi*k/6))^(n+1)) - Herbert Kociemba, Jun 02 2004

a(n)= sum{k=0..floor(n/2), binomial(n-k, k)2^(n-k)} - Paul Barry, Jul 13 2004

a(n) = A080040(n) - A028860(n+1) - Creighton Dement, Jan 19 2005

a(n)=Sum{k, 0<=k<=n}A112899(n,k) . - Philippe Deléham, Nov 21 2007

a(n)=Sum_{k, 0<=k<=n}A063967(n,k) . - Philippe Deléham, Nov 03 2006

a(n)=((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*sqrt(3)); a(n)=Sum{k=0..n, binomial(n, 2k+1)3^k}.

Binomial transform of expansion of sinh(sqrt(3)x)/sqrt(3) (0, 1, 0, 3, 0, 9, ...). E.g.f.: exp(x)*sinh(sqrt(3)*x)/sqrt(3). - Paul Barry, May 09 2003

a(n)=(1/3)*sum(k=1..5, sin(Pi*k/2)*sin(2*Pi*k/3)*(1+2*cos(Pi*k/6))^n) - Herbert Kociemba, Jun 02 2004

a(n+1)=((3+sqrt(3))*(1+sqrt(3))^n +(3-sqrt(3))*(1-sqrt(3))^n)/6 . [From Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009]

Antidiagonals sums of A081577. - J. M. Bergot, Dec 15 2012

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + 2*x)/( x*(4*k+4 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013

MAPLE

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+2*a[n-2]od: seq(a[n], n=0..33); # [From Zerinvary Lajos, Dec 15 2008]

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), b):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL2, ZL2, ZL2), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n)/3, n=2..31); - Zerinvary Lajos, Mar 08 2008

MATHEMATICA

Expand[Table[((1 + Sqrt[3])^n - (1 - Sqrt[3])^n)/(2Sqrt[3]), {n, 0, 30}]] - Artur Jasinski, Dec 10 2006

a[n_]:=(MatrixPower[{{1, 3}, {1, 1}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, -1, 40}] [From Vladimir Orlovsky, Feb 19 2010]

LinearRecurrence[{2, 2}, {0, 1}, 30] (* Robert G. Wilson v, Apr 13 2013 *)

PROG

(Sage) [lucas_number1(n, 2, -2) for n in xrange(0, 30)] # [From Zerinvary Lajos, Apr 22 2009]

(PARI) Vec(x/(1-2*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 10 2011

(MAGMA) [Floor(((1 + Sqrt(3))^n - (1 - Sqrt(3))^n)/(2*Sqrt(3))): n in [0..30]]; // Vincenzo Librandi, Aug 18 2011

(Haskell)

a002605 n = a002605_list !! n

a002605_list =

   0 : 1 : map (* 2) (zipWith (+) a002605_list (tail a002605_list))

-- Reinhard Zumkeller, Oct 15 2011

CROSSREFS

First differences are given by A026150.

a(n) = A073387(n, 0), n>=0 (first column of triangle).

Cf. A080953, A026150, A052948, A077846, A080040, A028859.

Equals (1/3) A083337. First differences of A077846. Pairwise sums of A028860 and abs(A077917).

a(n)=A028860(n)/2 apart from the initial terms. [From Philippe Deléham, Nov 19 2008]

Row sums of A081577 and row sums of triangle A156710.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Cf. A030195, A106435, A108898, A125145.

Cf. A175289 (Pisano periods)

Sequence in context: A003142 A118041 A105073 * A026134 A105696 A074413

Adjacent sequences:  A002602 A002603 A002604 * A002606 A002607 A002608

KEYWORD

nonn,easy

AUTHOR

Colin Mallows

EXTENSIONS

Edited by N. J. A. Sloane, Apr 15 2009

STATUS

approved

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Last modified April 19 21:10 EDT 2014. Contains 240777 sequences.