0,3

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 (kociemba(AT)t-online.de), 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 (hillcino368(AT)gmail.com), Sep 25 2005

The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007

[1,3; 1,1]^n *[1,0] = [A026150(n), A002605(n)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008

(1+sqrt(3))^n = A026150(n) + A002605(n)*sqrt(3) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008

a(n) 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 (critzer.geoffrey(AT)usd443), Feb 07 2009]

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

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 476

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to Chebyshev polynomials.

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) observes that a(n)=(-I*sqrt(2))^n*U(n, I/sqrt(2)), U(n, x) = Chebyshev U-polynomial.

G.f.: 1/(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 (pbarry(AT)wit.ie), 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 (kociemba(AT)t-online.de), Jun 02 2004

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

A002605(n) = A080040(n) - A028860(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 19 2005

a(n)=Sum{k, 0<=k<=n}A112899(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007

a(n)=Sum_{k, 0<=k<=n}A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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}; G.f.: x/(1-2x-2x^2)

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 (pbarry(AT)wit.ie), May 09 2003

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

a(n)=((3+sqrt3)(1+sqrt3)^n+(3-sqrt3)(1-sqrt3)^n)/6 offset 0. Add a leading 0. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009]

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 (zerinvarylajos(AT)yahoo.com), 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 (zerinvarylajos(AT)yahoo.com), Mar 08 2008

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

(Other) sage: [lucas_number1(n, 2, -2) for n in xrange(0, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

First differences are given by A026150.

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

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

Cf. A002605, A028859.

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

a(n)=A028860(n)/2 apart from the initial terms. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2008]

Row sums of Pascal-(1,2,1) triangle A081577. - Paul Barry (pbarry(AT)wit.ie), Jan 24 2005

Equals row sums of triangle A156710 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 14 2009]

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Sequence in context: A027068 A118041 A105073 this_sequence A026134 A105696 A074413

Adjacent sequences: A002602 A002603 A002604 this_sequence A002606 A002607 A002608

nonn

C. L. Mallows (colinm(AT)research.avayalabs.com)

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