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A007070
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a(n)=4a(n-1)-2a(n-2).
(Formerly M3482)
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39
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1, 4, 14, 48, 164, 560, 1912, 6528, 22288, 76096, 259808, 887040, 3028544, 10340096, 35303296, 120532992, 411525376, 1405035520, 4797091328, 16378294272, 55918994432, 190919389184, 651839567872
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 (simoseve(AT)gmail.com), 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]
For n>=1, a(n-1) is the number of generalized compositions of n when there are 2^i-1 different types of i, (i=1,2,...). [From Milan R. Janjic (agnus(AT)blic.net), Sep 24 2010]
Binomial transform of A078057. - R. J. Mathar, Mar 28 2011
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REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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LINKS
| 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.
Index to sequences with linear recurrences with constant coefficients, signature(4,-2).
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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FORMULA
| 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+1)-(2-sqrt(2))^(n+1))/sqrt(8). - Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008, corrected Mar 28 2011
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) = 3a(n - 1) + a(n - 2) + a(n - 3) + ... + a(0) + 1. [From Gary W. Adamson, Feb 18 2011]
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EXAMPLE
| a(3) = 48 = 3 * 4 + 4 + 1 + 1 = 3*a(2) + a(1) + a(0) + 1.
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MATHEMATICA
| a=1; b=0; c=0; lst={}; Do[c=a+b+c; b=a+c; AppendTo[lst, c]; a=b; b=c, {n, 2*4!}]; lst..and/or.. a=1; b=4; lst=Table[c=4*b-2*a; a=b; b=c, {n, 0, 2*4!}]; PrependTo[lst, 4]; PrependTo[lst, 1] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 21 2010]
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PROG
| (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]
(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-8); S:=[ ((4+r)^(1+n)-(4-r)^(1+n))/((2^(1+n))*r): n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Vincenzo Librandi, Mar 27 2011
(MAGMA) [n le 2 select 3*n-2 else 4*Self(n-1)-2*Self(n-2): n in [1..23]]; // Bruno Berselli, Mar 28 2011
(Haskell)
a007070 n = a007070_list !! n
a007070_list = 1 : 4 : (map (* 2) $ zipWith (-)
(tail $ map (* 2) a007070_list) a007070_list)
-- Reinhard Zumkeller, Jan 16 2012
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CROSSREFS
| 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 * A204089 A092489 A094827
Adjacent sequences: A007067 A007068 A007069 * A007071 A007072 A007073
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Simon Plouffe (simon.plouffe(AT)gmail.com)
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