A212697 a(n) = 2*n*3^(n-1).

2, 12, 54, 216, 810, 2916, 10206, 34992, 118098, 393660, 1299078, 4251528, 13817466, 44641044, 143489070, 459165024, 1463588514, 4649045868, 14721978582, 46490458680, 146444944842, 460255540932, 1443528742014, 4518872583696, 14121476824050, 44059007691036

Offset: 1

Author

Stanislav Sykora, May 24 2012

Keywords

• nonn
• easy

Comments

Main transitions in systems of n particles with spin 1.

Consider the set S of all b^n numbers which have n digits in base b. Define as "main transition" a pair (x,y) of elements of S such that x and y differ in base b in only one digit which in y exceeds that in x by 1. This particular sequence a(n) gives the number of such transitions for the case b=3.

The terminology originates from quantum theory of coupled spin systems (such as in magnetic resonance) with n particles, each with spin S = (b-1)/2. Then the i-th digit's value in base b can be intended as a label for the b = 2S+1 quantum states of the i-th particle. The most intense main quantum transitions then correspond to the above definition. Due to continuity, the correspondence holds regardless of how strongly coupled are the particles among themselves.

a(n) is the number of functions from {1,2,...,n} into {1,2,3} with a specially designated element of the domain that is restricted to be mapped into {1,2}. Hence the e.g.f. is 2*x*exp(x)^3. - Geoffrey Critzer, Mar 01 2015

a(n) is the distance spectral radius of the dimension-regular generalized recursive circulant graph (commonly known as multiplicative circulant graph) of order 3^n. - John Rafael M. Antalan, Sep 25 2020

Examples

			n=2, b=3, S={00, 01, 02, 10, 11, 12, 20, 21, 22}, main transitions = {(00,01), (00,10), (01,02), (01,12), (02,12), (10,11), (10,20), (11,12), (11,21), (12,22), (20,21), (21,22)}, main transitions count = 12.
		

References

• M. H. Levitt, Spin Dynamics, Basics of Nuclear Magnetic Resonance, 2nd Edition, John Wiley & Sons, 2007, Section 6 (Mathematical techniques).
• J. A. Pople, W. G. Schneider, H. J. Bernstein, High-Resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..100
• John Rafael M. Antalan and Francis Joseph H. Campeña, Distance eigenvalues and forwarding indices of dimension-regular generalized recursive circulant graph of order power of two and three, arXiv:2009.11608[math.CO], 2020.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
• Index entries for linear recurrences with constant coefficients, signature (6,-9).

Crossrefs

Cf. A001787 (b = 2).

Cf. A212698, A212699, A212700, A212701, A212702, A212703, A212704 (b = 4, 5, 6, 7, 8, 9, 10).

Row n=3 of A258997.

Programs

GAP

List([1..30], n-> 2*3^(n-1)*n) # G. C. Greubel, Jun 08 2019

Magma

[2*3^(n-1)*n: n in [1..30]]; // G. C. Greubel, Jun 08 2019

Maple

A212697:=n->2*n*3^(n-1): seq(A212697(n), n=1..30); # Wesley Ivan Hurt, Mar 01 2015

Mathematica

Table[Sum[Binomial[n, j] j 2^j, {j, n}], {n, 30}] (* Geoffrey Critzer, Mar 01 2015 *)
Table[2*3^(n-1)*n, {n,30}] (* G. C. Greubel, Jun 08 2019 *)

PARI

mtrans(n,b) = n*(b-1)*b^(n-1);
for (n=1,100,write("b212697.txt",n," ",mtrans(n,3)))

Sage

[2*3^(n-1)*n for n in (1..30)] # G. C. Greubel, Jun 08 2019

Formula

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=3.

From R. J. Mathar, Oct 15 2013: (Start)

G.f.: 2*x/(1-3*x)^2.

a(n) = 2*A027471(n+1). (End)

a(n) = A005843(n)*A000244(n-1). - Omar E. Pol, Jan 21 2014

a(n) = Sum_{j=1..n} binomial(n,j)*j*2^j. - Geoffrey Critzer, Mar 01 2015

E.g.f.: 2*x*exp(3*x). - G. C. Greubel, Jun 08 2019