0,2

a(n)/a(n-1) approaches 2+2^(1/2). Zak Seidov (zakseidov(AT)yahoo.com), Oct 12 2002

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

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

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

a(n) = sum of (n+1)-th row terms of triangle A140070. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 04 2008

The binomial transform is in A083878, the Catalan transform in A084868. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2008]

Equals row sums of triangle A152252 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Birkh\"{a}user, Boston, 3rd edition, 1990, p. 86.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.8 Answer to Exer. 8.

Index entries for sequences related to linear recurrences with constant coefficients

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 155

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

G.f.: (1-2x)/(1-4x+2x^2).

Binomial transform of A001333. E.g.f. exp(2x)cosh(x*sqrt(2)) - Paul Barry (pbarry(AT)wit.ie), May 08 2003

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

G.f.: (1-2x)/(1-4x+2x^2). a(n)=((2+sqrt(2))^n+(2-sqrt(2))^n)/2.

a(n) = Sum_{k, 0<=k<=n}2^k*A098158(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2006

a(n) = A007070(n)-2*A007070(n-1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

((2+sqrt2)^n+(2-sqrt2)^n)/2. The offset is 0. a(3)=20. - Al Hakanson (hawkuu(AT)gmail.com), Oct 15 2008

a(n)=Sum_{k, 0<=k<=n}A147703(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 29 2008]

A006012:=-(-1+2*z)/(1-4*z+2*z**2); [S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=if(n<0, 0, real(((2+quadgen(8))^n))) - Michael Somos Feb 12 2004

(PARI) a(n)=if(n<0, 0, polsym(x^2-4*x+2, n)[n+1]/2) - Michael Somos Feb 12 2004

(Other) sage: [lucas_number2(n, 4, 2)/2 for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]

a(n)=2*A007052(n-1)=A056236(n)/2.

Cf. A140070.

A152252 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2008]

Sequence in context: A148477 A027063 A027065 this_sequence A127152 A150120 A150121

Adjacent sequences: A006009 A006010 A006011 this_sequence A006013 A006014 A006015

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001