A014370 -id:A014370 - cp's OEIS frontend

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.

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, 241, 250, 258, 265, 271, 276, 280, 283, 285, 286

Offset: 1

Omar E. Pol, Jul 15 2012

			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;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

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

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

row =: monad define
    d=.>y
    < |. (+/d)-d
)
;}. row"0 <\ +/\ 1+i.11 NB. Vanessa McHale (vamchale(AT)gmail.com), Mar 01 2025

Accumulate[Flatten[Range[Range[15], 1, -1]]] (* Paolo Xausa, Mar 15 2025 *)

row(n) = vector(n, k, n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2); \\ Michel Marcus, Mar 10 2025

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
T(n,k) = n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2. - Davide Rotondo, Mar 10 2025
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

More terms from Michel Marcus, Mar 10 2025

A332662 Put-and-count: An enumeration of N X N where N = {0, 1, 2, ...}. The terms are interleaved x and y coordinates. Or: A row-wise storage scheme for sequences of regular triangles.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 1, 1, 2, 1, 3, 0, 4, 0, 5, 0, 0, 3, 1, 2, 2, 2, 3, 1, 4, 1, 5, 1, 6, 0, 7, 0, 8, 0, 9, 0, 0, 4, 1, 3, 2, 3, 3, 2, 4, 2, 5, 2, 6, 1, 7, 1, 8, 1, 9, 1, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 0, 5, 1, 4, 2, 4, 3, 3, 4, 3, 5, 3, 6, 2, 7, 2

Offset: 0

Peter Luschny, Feb 18 2020

nonn

Other enumerations of N X N designed with storage allocation for extensible arrays in mind include A319514 and A319571.

			Illustrating the linear storage layout of a sequence of regular triangles.
(A) [ 0], [ 2,  3], [ 7,  8,  9], [16, 17, 18, 19], [30, 31, 32, 33, 34], ...
(B) [ 1], [ 5,  6], [13, 14, 15], [26, 27, 28, 29], ...
(C) [ 4], [11, 12], [23, 24, 25], ...
(D) [10], [21, 22], ...
(E) [20], ...
...
The first column is A000292.
The start values of all partial rows (in ascending order) are 0 plus A014370.
The start values of the partial rows in the first row are A005581 (without first 0).
The start values of the partial rows on the main diagonal are A331987.
The end values of all partial rows (in ascending order) are A332023.
The end values of the partial rows in the first row are A062748.
The end values of the partial rows on the main diagonal are A332698.

Peter Luschny, Table of n, a(n) for n = 0..10000

A332663 (x-coordinates), A056559 (y-coordinates).
Cf. A000292, A014370, A002260, A005581, A319514, A319571, A331987.
Cf. A332023, A062748, A332698.

function a_list(N)
    a = Int[]
    for n in 1:N
        i = 0
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, i, t)
                i += 1
end end end; a end
a_list(5) |> println

count := (k, A) -> ListTools:-Occurrences(k, A): t := n -> n*(n+1)/2:
PutAndCount := proc(N) local L, n, v, c, seq; L := NULL; seq := NULL;
for n from 1 to N do
   for v from 0 to t(n)-1 do
     # How often did you see v in this sequence before?
     c := count(v, [seq]);
     L := L, v, c; seq := seq, v;
od od; L end:  PutAndCount(6);
# Returning 'seq' instead of 'L' gives the x-coordinates (A332663).

t[n_] := n*(n+1)/2;
PutAndCount[N_] := Module[{L, n, v, c, seq},
L = {}; seq = {};
For[n = 1, n <= N, n++,
   For[v = 0, v <= t[n]-1, v++,
      c = Count[seq, v];
      L = Join[L, {v, c}]; seq = Append[seq, v]
]]; L];
PutAndCount[6] (* Jean-François Alcover, Oct 13 2024, after Maple program *)

A109402 Partial sums of A109400.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 12, 14, 17, 19, 22, 27, 28, 30, 33, 37, 39, 42, 47, 54, 55, 57, 60, 64, 69, 71, 74, 79, 86, 97, 98, 100, 103, 107, 112, 118, 120, 123, 128, 135, 146, 159, 160, 162, 165, 169, 174, 180, 187, 189, 192, 197, 204, 215, 228, 245, 246

Offset: 1

Reinhard Zumkeller, Jun 27 2005

nonn

a(n) = Sum(A109400(m): 1<=m<=n).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007504, A014370.

Accumulate[With[{prs=Prime[Range[20]]},Flatten[Table[{Range[n],Take[prs,n]},{n,10}]]]] (* Harvey P. Dale, May 25 2025 *)

A332023 T(n, k) = binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 12, 15, 19, 20, 22, 25, 29, 34, 35, 37, 40, 44, 49, 55, 56, 58, 61, 65, 70, 76, 83, 84, 86, 89, 93, 98, 104, 111, 119, 120, 122, 125, 129, 134, 140, 147, 155, 164, 165, 167, 170, 174, 179, 185, 192, 200, 209, 219

Offset: 0

Peter Luschny, Feb 20 2020

The sequence increases monotonically.

			The triangle starts:
[0]   0;
[1]   1,   3;
[2]   4,   6,   9;
[3]  10,  12,  15,  19;
[4]  20,  22,  25,  29,  34;
[5]  35,  37,  40,  44,  49,  55;
[6]  56,  58,  61,  65,  70,  76,  83;
[7]  84,  86,  89,  93,  98, 104, 111, 119;
[8] 120, 122, 125, 129, 134, 140, 147, 155, 164;
[9] 165, 167, 170, 174, 179, 185, 192, 200, 209, 219;

Cf. A000292 (first column), A062748 (diagonal), A005286 (subdiagonal), A332697 (row sums).

Cf. A014370.

T := (n, k) -> binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1):
seq(seq(T(n, k), k=0..n), n=0..9);

T(n, k) = (1/6)*(3*k^2 + 9*k + n*(n + 1)*(n + 2)).

A332663 Even bisection of A332662: the x-coordinates of an enumeration of N X N.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

Offset: 0

Peter Luschny, Feb 19 2020

nonn

Cf. A332662, A002260, A014370, A000292 (positions of 0).

Maple
```
# See A332662.
```

cp's OEIS Frontend

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.

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A332662 Put-and-count: An enumeration of N X N where N = {0, 1, 2, ...}. The terms are interleaved x and y coordinates. Or: A row-wise storage scheme for sequences of regular triangles.

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A109402 Partial sums of A109400.

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A332023 T(n, k) = binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1). Triangle read by rows, T(n, k) for 0 <= k <= n.

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A332663 Even bisection of A332662: the x-coordinates of an enumeration of N X N.

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