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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..56.

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.

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

AUTHOR

Omar E. Pol, Jul 15 2012

STATUS

approved

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Last modified August 19 08:17 EDT 2019. Contains 326115 sequences. (Running on oeis4.)