A212013 - OEIS

%I #30 Jan 16 2018 19:06:52

%S 1,3,4,7,9,10,14,17,19,20,25,29,32,34,35,41,46,50,53,55,56,63,69,74,

%T 78,81,83,84,92,99,105,110,114,117,119,120,129,137,144,150,155,159,

%U 162,164,165,175,184,192,199,205,210,214,217,219,220,231

%N 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.

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

%F 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

%e Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:

%e     1;

%e     3,   4;

%e     7,   9,  10;

%e    14,  17,  19,  20;

%e    25,  29,  32,  34,  35;

%e    41,  46,  50,  53,  55,  56;

%e    63,  69,  74,  78,  81,  83,  84;

%e    92,  99, 105, 110, 114, 117, 119, 120;

%e   129, 137, 144, 150, 155, 159, 162, 164, 165;

%e   175, 184, 192, 199, 205, 210, 214, 217, 219, 220;

%e   ...

%e Column 1 gives positive terms of A004006. Right border gives positive terms of A000292. Row sums give positive terms of A006325.

%e 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:

%e     1;

%e     3,   4;

%e     7,   9,  10;

%e    14;

%e    17,  19,  20,  25;

%e    29,  32,  34,  35,  41;

%e    46,  50,  53,  55,  56,  63;

%e    69,  74,  78,  81,  83,  84,  92;

%e    99, 105, 110, 114, 117, 119, 120, 129;

%e   137, 144, 150, 155, 159, 162, 164, 165, 175;

%e   184, 192, 199, 205, 210, 214, 217, 219, 220, 231;

%e   ...

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

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

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Jul 15 2012