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A171842 Binomial transform of 1,0,1,0,2,0,4,0,8,0,16,... 2
1, 1, 2, 4, 9, 21, 50, 120, 289, 697, 1682, 4060, 9801, 23661, 57122, 137904, 332929, 803761, 1940450, 4684660, 11309769, 27304197, 65918162, 159140520, 384199201, 927538921, 2239277042, 5406093004, 13051463049, 31509019101, 76069501250, 183648021600, 443365544449, 1070379110497 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of nonisomorphic n-element interval orders with no 3-element antichain. - Richard Stanley, Nov 21 2011

a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 0, 1; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..800

Michael D. Barrus, Weakly threshold graphs, arXiv preprint arXiv:1608.01358 [math.CO], 2016.

Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv:1105.3713 [math.CO], 2011.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

FORMULA

a(n) = A024537(n-1), n>0. - R. J. Mathar, Jan 28 2010

a(n) = 3*a(n-1)-a(n-2)-a(n-3). G.f.: (1-2*x)/((1-x)*(1-2*x-x^2)). - Colin Barker, Apr 01 2012

a(n) = (2+(1-sqrt(2))^n+(1+sqrt(2))^n)/4. - Colin Barker, Mar 16 2016

MAPLE

read("transforms") :

L := [1, seq(2^i, i=0..30)] ;

AERATE(L, 1) ;

BINOMIAL(%) ; # R. J. Mathar, Sep 26 2011

MATHEMATICA

LinearRecurrence[{3, -1, -1}, {1, 1, 2}, 50] (* Jean-François Alcover, Feb 25 2017 *)

PROG

(PARI) Vec((1-2*x)/((1-x)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, Mar 16 2016

CROSSREFS

Sequence in context: A275864 A018905 A024537 * A027826 A261664 A091964

Adjacent sequences:  A171839 A171840 A171841 * A171843 A171844 A171845

KEYWORD

nonn,easy

AUTHOR

Philippe Deléham, Dec 19 2009

STATUS

approved

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Last modified November 24 00:27 EST 2017. Contains 295164 sequences.