A033547 Otto Haxel's guess for magic numbers of nuclear shells.

0, 2, 6, 14, 28, 50, 82, 126, 184, 258, 350, 462, 596, 754, 938, 1150, 1392, 1666, 1974, 2318, 2700, 3122, 3586, 4094, 4648, 5250, 5902, 6606, 7364, 8178, 9050, 9982, 10976, 12034, 13158, 14350, 15612, 16946, 18354, 19838, 21400, 23042, 24766, 26574, 28468

Offset

0,2

Comments

O. Haxel gave a construction procedure. The formulas are due to Wolfdieter Lang.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Otto Haxel, Die Entstehung des Schalenmodells der Atomkerne, Physikalische Blätter, Vol. 50, Iss. 4 (1994), pp. 339-341. (in German)

Otto Haxel et al., On the "Magic Numbers" in Nuclear Structure, Phys. Rev., 75 (1949), 1766.

Vladimir Ladma, Magic numbers.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = n*(n^2 + 5)/3.

G.f.: 2*x*(1 - x + x^2)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 05 2015

E.g.f.: x*(6 + 3*x + x^2)*exp(x)/3. - G. C. Greubel, Oct 12 2019

a(n) = A046127(n+1) - 2. - Jianing Song, Feb 03 2024

a(n) = A050407(n+1) + A341209(n-1) for n > 0. - Elmo R. Oliveira, May 22 2026

Maple

A033547:=n->n*(n^2+5)/3: seq(A033547(n), n=0..50); # Wesley Ivan Hurt, Apr 05 2015

Mathematica

Table[n(n^2+5)/3, {n, 0, 50}] (* Harvey P. Dale, Apr 07 2011 *)
(* Alternative: *)
LinearRecurrence[{4, -6, 4, -1}, {0, 2, 6, 14}, 50] (* Vincenzo Librandi, Apr 06 2015 *)

Prog

(Magma) [n*(n^2+5)/3 : n in [0..50]]; // Wesley Ivan Hurt, Apr 05 2015
(PARI) a(n)=n*(n^2+5)/3 \\ Charles R Greathouse IV, Jun 25 2017
(SageMath) [n*(n^2+5)/3 for n in range(50)] # G. C. Greubel, Oct 12 2019
(GAP) List([0..50], n-> n*(n^2+5)/3); # G. C. Greubel, Oct 12 2019

Crossrefs

Equals 2*A004006, partial sums of A014206, 2*(partial sums of A000124).

Cf. A018226, A046127, A050407, A341209.

Sequence in context: A161212 A256058 A294867 * A050531 A290699 A027083

Adjacent sequences: A033544 A033545 A033546 * A033548 A033549 A033550

Keyword

easy,nonn,nice

Author

Wolfdieter Lang

Status

approved