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%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
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