A212701 Main transitions in systems of n particles with spin 3.

6, 84, 882, 8232, 72030, 605052, 4941258, 39530064, 311299254, 2421216420, 18643366434, 142367525496, 1079620401678, 8138676874188, 61040076556410, 455765904954528, 3389758918099302, 25124095510618356, 185639150161791186, 1367867422244777160, 10053825553499112126

Offset

1,1

Comments

Please, refer to the general explanation in A212697.

This sequence is for base b=7 (see formula), corresponding to spin S=(b-1)/2=3.

Links

Stanislav Sykora, Table of n, a(n) for n = 1..100

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 (14,-49).

Formula

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=7.

From Colin Barker, Jun 16 2015: (Start)

a(n) = 14*a(n-1) - 49*a(n-2) for n > 2.

G.f.: 6*x/(7*x-1)^2. (End)

From Elmo R. Oliveira, May 14 2025: (Start)

E.g.f.: 6*x*exp(7*x).

a(n) = 6*A027473(n) = A008588(n)*A000420(n-1). (End)

From Amiram Eldar, Oct 27 2025: (Start)

Sum_{n>=1} 1/a(n) = (7/6)*log(7/6).

Sum_{n>=1} (-1)^(n+1)/a(n) = (7/6)*log(8/7). (End)

Mathematica

LinearRecurrence[{14, -49}, {6, 84}, 20] (* Harvey P. Dale, Aug 02 2016 *)

Prog

(PARI) mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212701.txt", n, " ", mtrans(n, 7)))
(PARI) Vec(6*x/(7*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

Crossrefs

Cf. A001787, A212697, A212698, A212699, A212700, A212702, A212703, A212704 (b = 2, 3, 4, 5, 6, 8, 9, 10).

Cf. A000420, A008588, A027473.

Sequence in context: A371471 A098982 A229455 * A248338 A144514 A244284

Adjacent sequences: A212698 A212699 A212700 * A212702 A212703 A212704

Keyword

nonn,easy

Author

Stanislav Sykora, May 25 2012

Status

approved