A322744 -id:A322744 - cp's OEIS frontend

A334922 Square array T(n,k) = ((3/2)nk + (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 8, 6, 5, 6, 8, 10, 10, 8, 6, 7, 9, 13, 12, 13, 9, 7, 8, 11, 15, 16, 16, 15, 11, 8, 9, 12, 18, 18, 21, 18, 18, 12, 9, 10, 14, 20, 22, 24, 24, 22, 20, 14, 10, 11, 15, 23, 24, 29, 27, 29, 24, 23, 15, 11

Offset: 1

David Lovler, May 16 2020

T(n,k) is commutative, associative, has identity element 1 and has 0. Also it is distributive except when an even number is partitioned into two odd numbers. Thus it has a multiplicative structure similar to that of A319929, A322630, A322744 and A327259 except that T(odd,odd) is not always odd, T(even,even) is not always even and T(odd,even) is not always even.

T(n,k) is in the same form as the supplementary arrays of A327263 called U(i;n,k). Here (and in A334923) i is being incremented by 1/2. When i is incremented by 1/4 or less, array values cease to be all integers, although all of the multiplication rules still hold.

			Array begins:
1   2   3   4   5   6   7   8   9  10 ...
2   3   5   6   8   9  11  12  14  15 ...
3   5   8  10  13  15  18  20  23  25 ...
4   6  10  12  16  18  22  24  28  30 ...
5   8  13  16  21  24  29  32  37  40 ...
6   9  15  18  24  27  33  36  42  45 ...
7  11  18  22  29  33  40  44  51  55 ...
8  12  20  24  32  36  44  48  56  60 ...
9  14  23  28  37  42  51  56  65  70 ...
10 15  25  30  40  45  55  60  70  75 ...
...

David Lovler, Table of n, a(n) for n = 1..465

Cf. A319929, A322630, A322744, A327259, A327263, A334923.

Table[Function[n, ((3/2)*n*k + (1/2)*If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]])/2][m - k + 1], {m, 11}, {k, m}] // Flatten (* Michael De Vlieger, Jun 23 2020 *)

T(n,k) = 3*floor(n/2)*floor(k/2) + A319929(n,k).
T(n,k) = (A322630(n,k) + n*k)/2.
T(n,k) = (A319929 + A322744(n,k))/2.
T(n,k) = 2*n*k - A334923(n,k).

A334923 Square array T(n,k) = ((5/2)nk - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 10, 10, 10, 5, 6, 12, 14, 14, 12, 6, 7, 15, 17, 20, 17, 15, 7, 8, 17, 21, 24, 24, 21, 17, 8, 9, 20, 24, 30, 29, 30, 24, 20, 9, 10, 22, 28, 34, 36, 36, 34, 28, 22, 10, 11, 25, 31, 40, 41, 45, 41, 40, 31, 25, 11

Offset: 1

David Lovler, May 25 2020

T(n,k) is in the same form as the supplementary arrays of A327263 called U(i;n,k). Here (and in A334922) i is being incremented by 1/2. When i is incremented by 1/4 or less, array values cease to be all integers, although all of the multiplication rules still hold.

			Array begins:
1   2   3   4   5   6   7   8   9  10 ...
2   5   7  10  12  15  17  20  22  25 ...
3   7  10  14  17  21  24  28  31  35 ...
4  10  14  20  24  30  34  40  44  50 ...
5  12  17  24  29  36  41  48  53  60 ...
6  15  21  30  36  45  51  60  66  75 ...
7  17  24  34  41  51  58  68  75  85 ...
8  20  28  40  48  60  68  80  88 100 ...
9  22  31  44  53  66  75  88  97 110 ...
10 25  35  50  60  75  85 100 110 125 ...
...

David Lovler, Table of n, a(n) for n = 1..465

Cf. A319929, A322630, A322744, A327259, A327263, A334922.

Table[Function[n, ((5/2)*n*k - (1/2)*If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]])/2][m - k + 1], {m, 11}, {k, m}] // Flatten (* Michael De Vlieger, Jun 23 2020 *)

T(n,k) = 5*floor(n/2)*floor(k/2) + A319929(n,k).
T(n,k) = (n*k + A322744(n,k))/2.
T(n,k) = (A322630(n,k) + A327259(n,k))/2.
T(n,k) = 2*n*k - A334922(n,k).

cp's OEIS Frontend

A334922 Square array T(n,k) = ((3/2)nk + (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

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A334923 Square array T(n,k) = ((5/2)nk - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

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cp's OEIS Frontend

A334922 Square array T(n,k) = ((3/2)*n*k + (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

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Mathematica

Formula

A334923 Square array T(n,k) = ((5/2)*n*k - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

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Mathematica

Formula

A334922 Square array T(n,k) = ((3/2)nk + (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

A334923 Square array T(n,k) = ((5/2)nk - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.