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A334922 Square array T(n,k) = ((3/2)*n*k + (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 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.

LINKS

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

FORMULA

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).

EXAMPLE

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 ...

...

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A085578 A350066 A135646 * A301851 A101646 A166079

Adjacent sequences:  A334919 A334920 A334921 * A334923 A334924 A334925

KEYWORD

nonn,tabl

AUTHOR

David Lovler, May 16 2020

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)