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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.
7
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
OFFSET
1,2
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
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;
...
CROSSREFS
Partial sums of A004736. Other versions are A210983, A212123, A213363, A213373.
Sequence in context: A002977 A024799 A240531 * A256761 A039579 A274794
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jul 15 2012
STATUS
approved