%I A006012 M1644
%S A006012 1,2,6,20,68,232,792,2704,9232,31520,107616,367424,1254464,4283008,
%T A006012 14623104,49926400,170459392,581984768,1987020288,6784111616,
%U A006012 23162405888,79081400320,270000789504,921840357376,3147359850496
%N A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.
%C A006012 a(n)/a(n-1) approaches 2+2^(1/2). Zak Seidov (zakseidov(AT)yahoo.com), 
               Oct 12 2002
%C A006012 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - 
               s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 4, s(2n) = 4. - Herbert Kociemba 
               (kociemba(AT)t-online.de), Jun 12 2004
%C A006012 a(k) = [M^k]_2,2, where M is the following 3 by 3 matrix: M = [1,1,1;
               1,2,1;1,1,1]. - Simone Severini (ss54(AT)york.ac.uk), Jun 11 2006
%C A006012 a(n-1) counts permutations pi on [n] for which the pairs {i, pi(i)} with 
               i < pi(i), considered as closed intervals [i+1,pi(i)], do not overlap; 
               equivalently, for each i in [n] there is at most one j <= i with 
               pi(j) > i. Counting these permutations by the position of n yields 
               the recurrence relation. - David Callan (callan(AT)stat.wisc.edu), 
               Sep 02 2003
%C A006012 a(n) = sum of (n+1)-th row terms of triangle A140070. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), May 04 2008
%C A006012 The binomial transform is in A083878, the Catalan transform in A084868. 
               [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2008]
%C A006012 Equals row sums of triangle A152252 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 30 2008]
%D A006012 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006012 D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. 
               Birkh\"{a}user, Boston, 3rd edition, 1990, p. 86.
%D A006012 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, 
               MA, Vol. 3, Sect 5.4.8 Answer to Exer. 8.
%H A006012 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006012 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=155">
               Encyclopedia of Combinatorial Structures 155</a>
%H A006012 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A006012 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A006012 G.f.: (1-2x)/(1-4x+2x^2).
%F A006012 Binomial transform of A001333. E.g.f. exp(2x)cosh(x*sqrt(2)) - Paul Barry 
               (pbarry(AT)wit.ie), May 08 2003
%F A006012 a(n)=sum{k=0..floor(n/2), C(n, 2k)2^(n-k) }=sum{k=0..n, C(n, k)2^(n-k/
               2)(1+(-1)^n)/2} - Paul Barry (pbarry(AT)wit.ie), Nov 22 2003
%F A006012 G.f.: (1-2x)/(1-4x+2x^2). a(n)=((2+sqrt(2))^n+(2-sqrt(2))^n)/2.
%F A006012 a(n) = Sum_{k, 0<=k<=n}2^k*A098158(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 04 2006
%F A006012 a(n) = A007070(n)-2*A007070(n-1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 16 2007
%F A006012 ((2+sqrt2)^n+(2-sqrt2)^n)/2. The offset is 0. a(3)=20. - Al Hakanson 
               (hawkuu(AT)gmail.com), Oct 15 2008
%F A006012 a(n)=Sum_{k, 0<=k<=n}A147703(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 29 2008]
%p A006012 A006012:=-(-1+2*z)/(1-4*z+2*z**2); [S. Plouffe in his 1992 dissertation.]
%o A006012 (PARI) a(n)=if(n<0,0,real(((2+quadgen(8))^n))) - Michael Somos Feb 12 
               2004
%o A006012 (PARI) a(n)=if(n<0,0,polsym(x^2-4*x+2,n)[n+1]/2) - Michael Somos Feb 
               12 2004
%o A006012 (Other) sage: [lucas_number2(n,4,2)/2 for n in xrange(0, 25)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
%Y A006012 a(n)=2*A007052(n-1)=A056236(n)/2.
%Y A006012 Cf. A140070.
%Y A006012 A152252 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2008]
%Y A006012 Sequence in context: A148477 A027063 A027065 this_sequence A127152 A150120 
               A150121
%Y A006012 Adjacent sequences: A006009 A006010 A006011 this_sequence A006013 A006014 
               A006015
%K A006012 nonn,easy
%O A006012 0,2
%A A006012 N. J. A. Sloane (njas(AT)research.att.com).
%E A006012 More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001