%I A078057
%S A078057 1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243,275807,665857,
%T A078057 1607521,3880899,9369319,22619537,54608393,131836323,318281039,768398401,
%U A078057 1855077841,4478554083,10812186007,26102926097,63018038201,152139002499
%N A078057 Expansion of (1+x)/(1-2*x-x^2).
%C A078057 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) ].
%C A078057 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
%C A078057 The number of Khalimsky-continuous functions with one fixed endpoint. 
               - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
%C A078057 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
%D A078057 A. Froehlich and M. J. Taylor, Algebraic Number Theory, Cambridge, 1991 
               (see p. 3).
%D A078057 Munarini, Emanuele, Combinatorial properties of the antichains of a garland. 
               Integers, 9 (2009), 353-374.
%D A078057 Shiva Samieinia, Digital straight line segments and curves. Licentiate 
               Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
%D A078057 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).
%D A078057 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]
%H A078057 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A078057 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A078057 Shiva Samieinia, <a href="http://www.math.su.se/reports/2007/6/">Home 
               Page</a>.
%F A078057 a(0)=1; a(1)=3; a(n) = 2*a(n-1) + a(n-2) - Wayne VanWeerthuizen (waynemv(AT)yahoo.com), 
               May 02 2004
%F A078057 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
%F A078057 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]
%F A078057 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]
%t A078057 Expand[Table[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2, {n, 1, 30}]] - Artur 
               Jasinski (grafix(AT)csl.pl), Dec 10 2006
%Y A078057 Essentially the same as A001333, which has many more references.
%Y A078057 Cf. A131887, A131935, A000129.
%Y A078057 Sequence in context: A077851 A089737 A001333 this_sequence A123335 A089742 
               A131721
%Y A078057 Adjacent sequences: A078054 A078055 A078056 this_sequence A078058 A078059 
               A078060
%K A078057 nonn
%O A078057 0,2
%A A078057 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002