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

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

1,2

Comments

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 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.

Links

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

Formula

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).

Example

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 ...
...

Mathematica

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 *)

Crossrefs

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

Sequence in context: A131901 A204000 A132071 * A061177 A129312 A115262

Adjacent sequences: A334920 A334921 A334922 * A334924 A334925 A334926

Keyword

nonn,tabl

Author

David Lovler, May 25 2020

Status

approved