0,2

Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 4) is "size of raises in pot-limit poker, one blind, maximum raising".

It appears that this sequence is the BinomialMean transform of A002315 - see A075271. - John W. Layman (layman(AT)math.vt.edu), Oct 02 2002

Number of (s(0), s(1), ..., s(2n+3)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+3, s(0) = 1, s(2n+3) = 4. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004

a(n) = number of unique matrix products in (A+B+C+D)^n where commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D. - Paul D. Hanna and Joshua Zucker, Feb 01 2006

The n-th term of the sequence is the entry (1,2) in the n-th power of the matrix M=[1,-1;-1,3]. - Simone Severini (ss54(AT)york.ac.uk), Feb 15 2006

Hankel transform of this sequence is [1,-2,0,0,0,0,0,0,0,0,0,...]. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008: (Start)

a(n) = (n+1) terms of (1,1,4,14,48,...) convolved with (1,3,7,15,31,...);

e.g. a(4) = 164 = (1*31 + 1*15 + 4*7 + 14*3 + 48*1) = (31 + 15 + 28 + 42 + 48). (End)

Equals INVERT transform of A000225: (1, 3, 7, 15, 31,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 03 2009]

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

A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, preprint, 2007.

A. Fraenkel and C. Kimberling, "Generalized Wythoff arrays, shuffles and interspersions," Discrete Mathematics 126 (1994) 137-149.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Index entries for sequences related to linear recurrences with constant coefficients

A. Burstein, S. Kitaev and T. Mansour, Independent sets in certain classes of (almost) regular graphs

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 440

M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Index entries for sequences related to poker

Index entries for sequences related to Chebyshev polynomials.

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

Preceded by 0, this is the binomial transform of the Pell numbers A000129. Its E.g.f. is then exp(2x)sinh(sqrt(2)x)/sqrt(2). - Paul Barry (pbarry(AT)wit.ie), May 09 2003

a(n) = ((2+sqrt(2))^n-(2-sqrt(2))^n)/sqrt(8). - Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008

a(n) = (2-sqrt(2))^n*(1/2-sqrt(2)/2)+(2+sqrt(2))^n*(1/2+sqrt(2)/2) - Paul Barry (pbarry(AT)wit.ie), May 09 2003.

a(n)=ceil((2+sqrt(2))*a(n-1)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 15 2003

a(n)=U(n, sqrt(2))sqrt(2)^n - Paul Barry (pbarry(AT)wit.ie), Nov 19 2003

a(n)=(1/4)*Sum(r, 1, 7, Sin(r*Pi/8)Sin(r*Pi/2)(2Cos(r*Pi/8))^(2n+3)) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004

a(n) = center term in M^n * [1 1 1], where M = the 3X3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [A007052(n) a(n) A007052(n)]. E.g. a(3) = 48 since M^3 * [1 1 1] = [34 48 34], where 34 = A007052(3). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004

This is the binomial mean transform of A002307. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006

a(2n)=Sum(r,0,n,2^(2n-1-r)*(4*Binomial(2n-1,2r)+3*Binomial(2n-1,2r+1)) a(2n-1)=Sum(r,0,n,2^(2n-2-r)*(4*Binomial(2n-2,2r)+3*Binomial(2n-2,2r+1)) - Jeffrey E. Liese (jliese(AT)math.ucsd.edu), Oct 12 2006

a(n) = sum of row (n+1) of triangle A140071. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 04 2008

a(n)=((sqrt(2)+1)^n - (sqrt(2)-1)^n)) / 2. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Aug 06 2009]

(PARI) a(n)=polcoeff(1/(1-4*x+2*x^2)+x*O(x^n), n)

(PARI) a(n)=if(n<1, 1, ceil((2+sqrt(2))*a(n-1)))

(Other) sage: [lucas_number1(n, 4, 2) for n in xrange(1, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

Row sums of A059474. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 14 2006

Cf. A007052, A006012 (same recurrence).

Equals 2 * A003480, n>0.

Cf. A007052.

Cf. A140071.

Sequence in context: A027906 A047135 A127359 this_sequence A092489 A094827 A094667

Adjacent sequences: A007067 A007068 A007069 this_sequence A007071 A007072 A007073

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Simon Plouffe (simon.plouffe(AT)gmail.com)