%I #68 Sep 08 2022 08:46:02

%S 2,12,54,216,810,2916,10206,34992,118098,393660,1299078,4251528,

%T 13817466,44641044,143489070,459165024,1463588514,4649045868,

%U 14721978582,46490458680,146444944842,460255540932,1443528742014,4518872583696,14121476824050,44059007691036

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

%C Main transitions in systems of n particles with spin 1.

%C 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.

%C 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.

%C 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

%C 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

%D M. H. Levitt, Spin Dynamics, Basics of Nuclear Magnetic Resonance, 2nd Edition, John Wiley & Sons, 2007, Section 6 (Mathematical techniques).

%D J. A. Pople, W. G. Schneider, H. J. Bernstein, High-Resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

%H Stanislav Sykora, <a href="/A212697/b212697.txt">Table of n, a(n) for n = 1..100</a>

%H John Rafael M. Antalan and Francis Joseph H. Campeña, <a href="https://arxiv.org/abs/2009.11608">Distance eigenvalues and forwarding indices of dimension-regular generalized recursive circulant graph of order power of two and three</a>, arXiv:2009.11608[math.CO], 2020.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).

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

%F From _R. J. Mathar_, Oct 15 2013: (Start)

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

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

%F a(n) = A005843(n)*A000244(n-1). - _Omar E. Pol_, Jan 21 2014

%F a(n) = Sum_{j=1..n} binomial(n,j)*j*2^j. - _Geoffrey Critzer_, Mar 01 2015

%F E.g.f.: 2*x*exp(3*x). - _G. C. Greubel_, Jun 08 2019

%e 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.

%p A212697:=n->2*n*3^(n-1): seq(A212697(n), n=1..30); # _Wesley Ivan Hurt_, Mar 01 2015

%t Table[Sum[Binomial[n, j] j 2^j, {j, n}], {n, 30}] (* _Geoffrey Critzer_, Mar 01 2015 *)

%t Table[2*3^(n-1)*n, {n,30}] (* _G. C. Greubel_, Jun 08 2019 *)

%o (PARI) mtrans(n,b) = n*(b-1)*b^(n-1);

%o for (n=1,100,write("b212697.txt",n," ",mtrans(n,3)))

%o (Magma) [2*3^(n-1)*n: n in [1..30]]; // _G. C. Greubel_, Jun 08 2019

%o (Sage) [2*3^(n-1)*n for n in (1..30)] # _G. C. Greubel_, Jun 08 2019

%o (GAP) List([1..30], n-> 2*3^(n-1)*n) # _G. C. Greubel_, Jun 08 2019

%Y Cf. A001787 (b = 2).

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

%Y Row n=3 of A258997.

%K nonn,easy

%O 1,1

%A _Stanislav Sykora_, May 24 2012