0,2

Let x_n be the sequence 1,3,7,17,41,99,239,... (this sequence or A001333) and let y_n = 1,2,5,12,29,70,169,... (A000129). Then {+- x_n +- y_n*sqrt(2) } are the units in the ring of algebraic integers Z[ sqrt(2) ].

Consider a string of n red, blue and green beads (with start and end points distinct and not interchangeable). If one pairing is disallowed, so that a red bead cannot immediately follow a blue bead or vice versa, how many different strings exist of any given length? Answer is a(n). E.g. a(3)=17 because there are 17 strings of length 3: RRR, RRG, RGR, RGG, RGB, GRR, GRG, GGR, GGG, GGB, GBG, GBB, BGR, BGG, BGB, BBG, BBB - Wayne VanWeerthuizen (waynemv(AT)yahoo.com), May 02 2004

The number of Khalimsky-continuous functions with one fixed endpoint. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007

The sequence (-1)^C(n+1,2)*a(n) with g.f. (1-3x-x^2-x^3)/(1+6x^2+x^4) is the Hankel transform of the signed central binomial coefficients (-1)^C(n+1,2)*A001405(n). - Paul Barry (pbarry(AT)wit.ie), Jun 24 2008

A. Froehlich and M. J. Taylor, Algebraic Number Theory, Cambridge, 1991 (see p. 3).

Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.

Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.

Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).

Emanule Munarini, "Combinatorial properties of the antichains of a garland", INTEGERS, 9 (2009) 353-374. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 22 2009]

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to linear recurrences with constant coefficients

Shiva Samieinia, Home Page.

a(0)=1; a(1)=3; a(n) = 2*a(n-1) + a(n-2) - Wayne VanWeerthuizen (waynemv(AT)yahoo.com), May 02 2004

a(n) = 2*a(n-1) + a(n-2); a(n+1)/a(n) tends to silver ratio 1+\sqrt(2) as n tends to infinity. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007

a(n)=Sum_{k, 0<=k<=n}A147720(n,k)*3^k*(-1/3)^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 15 2008]

a(n)=(1/2)*[1+sqrt(2)]^n-(1/2)*sqrt(2)*[1-sqrt(2)]^n+(1/2)*[1-sqrt(2)]^n+(1/2)*[1+sqrt(2)]^n *sqrt(2), with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]

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

Essentially the same as A001333, which has many more references.

Cf. A131887, A131935, A000129.

Sequence in context: A077851 A089737 A001333 this_sequence A123335 A089742 A131721

Adjacent sequences: A078054 A078055 A078056 this_sequence A078058 A078059 A078060

nonn

N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002