A227712 a(n) = 9*2^n - 3*n - 5.

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

Offset: 0

Author

Emeric Deutsch, Aug 06 2013

Keywords

• nonn
• easy

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[n-1] 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.

Examples

			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.
		

Links

• R. Kopelman, M. Shortreed, Z. Y. Shi, W. Tan, Z. F. Xu, J. S. Moore, A. Bar-Haim, J. Klafter, Spectroscopic evidence for excitonic localization in fractal antenna supermolecules, Phys. Rev. Letters, 78, 1997, 1239-1242.
• M. A. Martín-Delgado, J. Rodriguez-Laguna, G. Sierra, A density matrix renormalization group study of excitons in dendrimers, Phys. Rev. B 65, 2002, 155116(1-11).
• S. Raychaudhuri, Y. Shapir, S. Mukamel, Disorder and funneling effect on exciton migration in treelike dendrimers, Phys. Rev. E, 65, 2002, 021803(1-12).
• S. Tretiak, V. Chernyak, S. Mukamel, Localized electronic excitations in phenylacetylene dendrimers, J. Phys. Chem. B, 102, 1998, 3310-3315.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A079583.

Programs

Magma

[9*2^n-3*n-5: n in [0..40]]; // Vincenzo Librandi, Feb 19 2016

Maple

a := proc (n) options operator, arrow: 9*2^n-3*n-5 end proc: seq(a(n), n = 0 .. 35);

Mathematica

Table[9*2^n-3n-5,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{4,10,25},40] (* Harvey P. Dale, Apr 15 2015 *)

PARI

Vec((4-6*x+5*x^2)/((1-2*x)*(1-x)^2) + O(x^100)) \\ Altug Alkan, Oct 17 2015

Formula

G.f.: (4-6*x+5*x^2)/((1-2*x)*(1-x)^2).

a(0)=4, a(1)=10, a(2)=25, a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - Harvey P. Dale, Apr 15 2015

a(n)= 3*A079583(n) + 1. - Emeric Deutsch, Feb 18 2016