

A123620


Expansion of (1 + x + x^2) / (1  3*x  3*x^2).


5



1, 4, 16, 60, 228, 864, 3276, 12420, 47088, 178524, 676836, 2566080, 9728748, 36884484, 139839696, 530172540, 2010036708, 7620627744, 28891993356, 109537863300, 415289569968, 1574482299804, 5969315609316, 22631393727360, 85802128010028, 325300565212164
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OFFSET

0,2


COMMENTS

From Johannes W. Meijer, Aug 14 2010: (Start)
A berserker sequence, see A180141. For the corner squares 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the side squares to A180142 and for the central square to A155116.
This sequence belongs to a family of sequences with GF(x) = (1+x+k*x^2)/(13*x+(k4)*x^2). Berserker sequences that are members of this family are 4*A055099(n) (k=2; with leading 1 added), A123620 (k=1; this sequence), A000302 (k=0), 4*A179606 (k=1; with leading 1 added) and A180141 (k=2). Some other members of this family are 4*A003688 (k=3; with leading 1 added), 4*A003946 (k=4; with leading 1 added), 4*A002878 (k=5; with leading 1 added) and 4*A033484 (k=6; with leading 1 added).
(End)
a(n) is the number of length n sequences on an alphabet of 4 letters that do not contain more than 2 consecutive equal letters. For example, a(3)=60 because we count all 4^3=64 words except: aaa, bbb, ccc, ddd.  Geoffrey Critzer, Mar 12 2014


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Burstein and T. Mansour, Words restricted by 3letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
A. Burstein and T. Mansour, Words restricted by 3letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 114.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the nanacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 205
Index entries for linear recurrences with constant coefficients, signature (3,3).


FORMULA

a(0)=1, a(1)=4, a(2)=16, a(n)=3*a(n1)+3*a(n2) for n>2.  Philippe Deléham, Sep 18 2009
a(n) = ((2^(1n)*((3sqrt(21))^(1+n) + (3+sqrt(21))^(1+n)))) / (3*sqrt(21)) for n>0.  Colin Barker, Oct 17 2017


MATHEMATICA

nn=25; CoefficientList[Series[(1z^(m+1))/(1r z +(r1)z^(m+1))/.{r>4, m>2}, {z, 0, nn}], z] (* Geoffrey Critzer, Mar 12 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1  3 x  3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)


PROG

(PARI) x='x+O('x^50); Vec((1+x+x^2)/(13*x3*x^2)) \\ G. C. Greubel, Oct 16 2017 *)
(MAGMA) [1] cat [Round(((2^(1n)*((3Sqrt(21))^(1+n) + (3+Sqrt(21))^(1+n))))/(3*Sqrt(21))): n in [1..50]]; // G. C. Greubel, Oct 26 2017


CROSSREFS

Column 4 in A265584.
Sequence in context: A269462 A047097 A051043 * A234008 A203153 A126929
Adjacent sequences: A123617 A123618 A123619 * A123621 A123622 A123623


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Nov 20 2006


STATUS

approved



