0,2

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

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

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

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.

Tanya Khovanova, Recursive Sequences

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 476

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

a(n)=((1+Sqrt(3))^(n+1)-(1-Sqrt(3))^(n+1))/(2Sqrt(3)). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 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}A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006

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

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=3..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008

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

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

Essentially the same as A080953.

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

Cf. A026150.

Adjacent sequences: A002602 A002603 A002604 this_sequence A002606 A002607 A002608

Sequence in context: A027068 A118041 A105073 this_sequence A080953 A026134 A105696

nonn

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