A212013 Triangle read by rows: total number of pairs of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

1, 3, 4, 7, 9, 10, 14, 17, 19, 20, 25, 29, 32, 34, 35, 41, 46, 50, 53, 55, 56, 63, 69, 74, 78, 81, 83, 84, 92, 99, 105, 110, 114, 117, 119, 120, 129, 137, 144, 150, 155, 159, 162, 164, 165, 175, 184, 192, 199, 205, 210, 214, 217, 219, 220, 231, 241, 250, 258, 265, 271, 276, 280, 283, 285, 286

Offset

1,2

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Formula

a(n) = A212014(n)/2.

Let R = floor(sqrt(8*n+1)) and S = floor(R/2) + R mod 2; then a(n) = binomial(S,3) + n + (n-binomial(S,2))*(S*(S+3)-2*n-2)/4. - Gerald Hillier, Jan 16 2018

T(n,k) = n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2. - Davide Rotondo, Mar 10 2025

G.f.: x*y*(1 - x + x^2*(1 - 3*y) - x^5*y^3 + x^3*y*(1 + y) - x^4*y*(1 - 2*y))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Mar 10 2025

Example

Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:
    1;
    3,   4;
    7,   9,  10;
   14,  17,  19,  20;
   25,  29,  32,  34,  35;
   41,  46,  50,  53,  55,  56;
   63,  69,  74,  78,  81,  83,  84;
   92,  99, 105, 110, 114, 117, 119, 120;
  129, 137, 144, 150, 155, 159, 162, 164, 165;
  175, 184, 192, 199, 205, 210, 214, 217, 219, 220;
  ...
Column 1 gives positive terms of A004006. Right border gives positive terms of A000292. Row sums give positive terms of A006325.
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. Note that in this case row 4 has only one term. Triangle begins:
    1;
    3,   4;
    7,   9,  10;
   14;
   17,  19,  20,  25;
   29,  32,  34,  35,  41;
   46,  50,  53,  55,  56,  63;
   69,  74,  78,  81,  83,  84,  92;
   99, 105, 110, 114, 117, 119, 120, 129;
  137, 144, 150, 155, 159, 162, 164, 165, 175;
  184, 192, 199, 205, 210, 214, 217, 219, 220, 231;
  ...

Mathematica

Accumulate[Flatten[Range[Range[15], 1, -1]]] (* Paolo Xausa, Mar 15 2025 *)

Prog

(J)
row =: monad define
    d=.>y
    < |. (+/d)-d
)
; }. row"0 <\ +/\ 1+i.11 NB. Vanessa McHale (vamchale(AT)gmail.com), Mar 01 2025
(PARI) row(n) = vector(n, k, n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2); \\ Michel Marcus, Mar 10 2025

Crossrefs

Partial sums of A004736. Other versions are A210983, A212123, A213363, A213373.

Cf. A000292, A004006, A006325, A212012, A212014, A014370.

Sequence in context: A002977 A024799 A240531 * A256761 A039579 A274794

Adjacent sequences: A212010 A212011 A212012 * A212014 A212015 A212016

Keyword

nonn,tabl,easy,changed

Author

Omar E. Pol, Jul 15 2012

Extensions

More terms from Michel Marcus, Mar 10 2025

Status

approved