

A227712


a(n) = 9*2^n  3*n  5.


1



4, 10, 25, 58, 127, 268, 553, 1126, 2275, 4576, 9181, 18394, 36823, 73684, 147409, 294862, 589771, 1179592, 2359237, 4718530, 9437119, 18874300, 37748665, 75497398, 150994867, 301989808, 603979693, 1207959466, 2415919015, 4831838116, 9663676321, 19327352734
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OFFSET

0,1


COMMENTS

Denoting by P[n] the path on n vertices, a(n) is the number of vertices of the tree obtained by identifying the roots of 3 identical rooted trees g[n], where g[n] is obtained recursively in the following manner: g[0]=P[2] and g[n] (n>=1) is obtained by identifying the roots of 2 copies of g[n1] and one of the extremities of P[n+1]; the root of g[n] is defined to be the other extremity of P[n+1]. Most references contain pictures of these trees; however, the small circles have to be viewed as vertices rather than hexagons.


LINKS

Table of n, a(n) for n=0..31.
R. Kopelman, M. Shortreed, Z. Y. Shi, W. Tan, Z. F. Xu, J. S. Moore, A. BarHaim, J. Klafter, Spectroscopic evidence for excitonic localization in fractal antenna supermolecules, Phys. Rev. Letters, 78, 1997, 12391242.
M. A. MartÃnDelgado, J. RodriguezLaguna, G. Sierra, A density matrix renormalization group study of excitons in dendrimers, Phys. Rev. B 65, 2002, 155116(111).
S. Raychaudhuri, Y. Shapir, S. Mukamel, Disorder and funneling effect on exciton migration in treelike dendrimers, Phys. Rev. E, 65, 2002, 021803(112).
S. Tretiak, V. Chernyak, S. Mukamel, Localized electronic excitations in phenylacetylene dendrimers, J. Phys. Chem. B, 102, 1998, 33103315.
Index entries for linear recurrences with constant coefficients, signature (4,5,2).


FORMULA

G.f.: (46*x+5*x^2)/((12*x)*(1x)^2).
a(0)=4, a(1)=10, a(2)=25, a(n) = 4*a(n1)5*a(n2)+2*a(n3).  Harvey P. Dale, Apr 15 2015
a(n)= 3*A079583(n) + 1.  Emeric Deutsch, Feb 18 2016


EXAMPLE

a(1) = 10 because g[1] is the rooted tree in the shape of Y (4 vertices) and a "bouquet" of three Y's has 3*4  2 = 10 vertices.


MAPLE

a := proc (n) options operator, arrow: 9*2^n3*n5 end proc: seq(a(n), n = 0 .. 35);


MATHEMATICA

Table[9*2^n3n5, {n, 0, 40}] (* or *) LinearRecurrence[{4, 5, 2}, {4, 10, 25}, 40] (* Harvey P. Dale, Apr 15 2015 *)


PROG

(PARI) Vec((46*x+5*x^2)/((12*x)*(1x)^2) + O(x^100)) \\ Altug Alkan, Oct 17 2015
(MAGMA) [9*2^n3*n5: n in [0..40]]; // Vincenzo Librandi, Feb 19 2016


CROSSREFS

Cf. A079583.
Sequence in context: A300760 A229916 A113412 * A159297 A248731 A279101
Adjacent sequences: A227709 A227710 A227711 * A227713 A227714 A227715


KEYWORD

nonn,easy


AUTHOR

Emeric Deutsch, Aug 06 2013


STATUS

approved



