A212703 Main transitions in systems of n particles with spin 4.

8, 144, 1944, 23328, 262440, 2834352, 29760696, 306110016, 3099363912, 30993639120, 306837027288, 3012581722464, 29372671794024, 284688972772848, 2745215094595320, 26354064908115072, 252010745683850376, 2401514164751985936, 22814384565143866392, 216136274827678734240

Offset

1,1

Comments

Please, refer to the general explanation in A212697.

This sequence is for base b=9 (see formula), corresponding to spin S=(b-1)/2=4.

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 (18,-81).

Formula

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=9.

From Colin Barker, Jun 16 2015: (Start)

a(n) = 18*a(n-1) - 81*a(n-2) for n > 2.

G.f.: 8*x/(9*x-1)^2. (End)

From Elmo R. Oliveira, May 13 2025: (Start)

E.g.f.: 8*x*exp(9*x).

a(n) = 8*A053540(n) = A008590(n)*A001019(n-1). (End)

From Amiram Eldar, Oct 27 2025: (Start)

Sum_{n>=1} 1/a(n) = (9/8)*log(9/8).

Sum_{n>=1} (-1)^(n+1)/a(n) = (9/8)*log(10/9). (End)

Mathematica

LinearRecurrence[{18, -81}, {8, 144}, 30] (* Harvey P. Dale, Jun 28 2017 *)

Prog

(PARI) mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212703.txt", n, " ", mtrans(n, 9)))
(PARI) Vec(8*x/(9*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015
(PARI) a(n)=8*n*9^(n-1) \\ Charles R Greathouse IV, Jun 16 2015

Crossrefs

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212704 (b = 2, 3, 4, 5, 6, 7, 8, 10).

Cf. A001019, A008590, A053540.

Sequence in context: A241229 A274730 A263604 * A065409 A061899 A134492

Adjacent sequences: A212700 A212701 A212702 * A212704 A212705 A212706

Keyword

nonn,easy

Author

Stanislav Sykora, May 25 2012

Status

approved