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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
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
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
KEYWORD
nonn,tabl
AUTHOR
David Lovler, May 16 2020
STATUS
approved