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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.

%I A212013 #63 Mar 15 2025 16:36:04
%S A212013 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 A212013 78,81,83,84,92,99,105,110,114,117,119,120,129,137,144,150,155,159,
%U A212013 162,164,165,175,184,192,199,205,210,214,217,219,220,231,241,250,258,265,271,276,280,283,285,286
%N 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.
%H A212013 Paolo Xausa, <a href="/A212013/b212013.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened).
%F A212013 a(n) = A212014(n)/2.
%F A212013 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
%F A212013 T(n,k) = n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2. - _Davide Rotondo_, Mar 10 2025
%F A212013 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
%e A212013 Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:
%e A212013     1;
%e A212013     3,   4;
%e A212013     7,   9,  10;
%e A212013    14,  17,  19,  20;
%e A212013    25,  29,  32,  34,  35;
%e A212013    41,  46,  50,  53,  55,  56;
%e A212013    63,  69,  74,  78,  81,  83,  84;
%e A212013    92,  99, 105, 110, 114, 117, 119, 120;
%e A212013   129, 137, 144, 150, 155, 159, 162, 164, 165;
%e A212013   175, 184, 192, 199, 205, 210, 214, 217, 219, 220;
%e A212013   ...
%e A212013 Column 1 gives positive terms of A004006. Right border gives positive terms of A000292. Row sums give positive terms of A006325.
%e A212013 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 A212013     1;
%e A212013     3,   4;
%e A212013     7,   9,  10;
%e A212013    14;
%e A212013    17,  19,  20,  25;
%e A212013    29,  32,  34,  35,  41;
%e A212013    46,  50,  53,  55,  56,  63;
%e A212013    69,  74,  78,  81,  83,  84,  92;
%e A212013    99, 105, 110, 114, 117, 119, 120, 129;
%e A212013   137, 144, 150, 155, 159, 162, 164, 165, 175;
%e A212013   184, 192, 199, 205, 210, 214, 217, 219, 220, 231;
%e A212013   ...
%t A212013 Accumulate[Flatten[Range[Range[15], 1, -1]]] (* _Paolo Xausa_, Mar 15 2025 *)
%o A212013 (J)
%o A212013 row =: monad define
%o A212013     d=.>y
%o A212013     < |. (+/d)-d
%o A212013 )
%o A212013 ;}. row"0 <\ +/\ 1+i.11 NB. Vanessa McHale (vamchale(AT)gmail.com), Mar 01 2025
%o A212013 (PARI) row(n) = vector(n, k, n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2); \\ _Michel Marcus_, Mar 10 2025
%Y A212013 Partial sums of A004736. Other versions are A210983, A212123, A213363, A213373.
%Y A212013 Cf. A000292, A004006, A006325, A212012, A212014, A014370.
%K A212013 nonn,tabl,easy
%O A212013 1,2
%A A212013 _Omar E. Pol_, Jul 15 2012
%E A212013 More terms from _Michel Marcus_, Mar 10 2025