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A212697
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a(n) = 2*n*3^(n-1).
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12
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) = n*(b-1)*b^(n-1). For this sequence, set b=3.
G.f.: 2*x/(1-3*x)^2.
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EXAMPLE
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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.
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MAPLE
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MATHEMATICA
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Table[Sum[Binomial[n, j] j 2^j, {j, n}], {n, 30}] (* Geoffrey Critzer, Mar 01 2015 *)
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PROG
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(PARI) mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212697.txt", n, " ", mtrans(n, 3)))
(Magma) [2*3^(n-1)*n: n in [1..30]]; // G. C. Greubel, Jun 08 2019
(Sage) [2*3^(n-1)*n for n in (1..30)] # G. C. Greubel, Jun 08 2019
(GAP) List([1..30], n-> 2*3^(n-1)*n) # G. C. Greubel, Jun 08 2019
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CROSSREFS
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Cf. A212698, A212699, A212700, A212701, A212702, A212703, A212704 (b = 4, 5, 6, 7, 8, 9, 10).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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