A327259 Array T(n,k) = 2*n*k - A319929(n,k), n >= 1, k >= 1, read by antidiagonals.

1, 2, 2, 3, 8, 3, 4, 10, 10, 4, 5, 16, 13, 16, 5, 6, 18, 20, 20, 18, 6, 7, 24, 23, 32, 23, 24, 7, 8, 26, 30, 36, 36, 30, 26, 8, 9, 32, 33, 48, 41, 48, 33, 32, 9, 10, 34, 40, 52, 54, 54, 52, 40, 34, 10, 11, 40, 43, 64, 59, 72, 59, 64, 43, 40, 11, 12, 42, 50, 68, 72, 78, 78, 72, 68, 50, 42, 12

Offset

1,2

Comments

Associative multiplication-like table whose values depend on whether n and k are odd or even.

Associativity is proved by checking the formula with eight cases of three odd and even arguments. T(n,k) is distributive as long as partitioning an even number into two odd numbers is not allowed.

T(n,k) has the same group structure as A319929, A322630 and A322744. For those arrays, position (3,3) is 5, 7 and 11 respectively. T(3,3) = 13. If we didn't have the formula for these arrays, their entries could be computed knowing one position and applying the arithmetic rules.

Links

David Lovler, Table of n, a(n) for n = 1..465 [restored by Georg Fischer, Oct 14 2019]

Formula

T(n,k) = 2*n*k - n - k + 1 if n is odd and k is odd;

T(n,k) = 2*n*k - n if n is even and k is odd;

T(n,k) = 2*n*k - k if n is odd and k is even;

T(n,k) = 2*n*k if n is even and k is even.

T(n,k) = 8*floor(n/2)*floor(k/2) + A319929(n,k).

T(n,n) = A354595(n). - David Lovler, Jul 09 2022

Writing T(n,k) as (4*n*k - 2*A319929(n,k))/2 shows that the array is U(4;n,k) of A327263. - David Lovler, Jan 15 2022

Example

Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   8  10  16  18  24  26  32  34  40
   3  10  13  20  23  30  33  40  43  50
   4  16  20  32  36  48  52  64  68  80
   5  18  23  36  41  54  59  72  77  90
   6  24  30  48  54  72  78  96 102 120
   7  26  33  52  59  78  85 104 111 130
   8  32  40  64  72  96 104 128 136 160
   9  34  43  68  77 102 111 136 145 170
  10  40  50  80  90 120 130 160 170 200

Mathematica

T[n_, k_]:=2n*k-If[Mod[n, 2]==1, If[Mod[k, 2]==1, n+k-1, k], If[Mod[k, 2]==1, n, 0]]; MatrixForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]] (* Stefano Spezia, Sep 05 2019 *)

Prog

(PARI) T(n, k) = 2*n*k - if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
matrix(8, 8, n, k, T(n, k)) \\ Michel Marcus, Sep 04 2019

Crossrefs

Equals A322744 + A322630 - A319929.

Equals 4*A322630 - 3*A319929.

Cf. A327260, A327261, A327263.

0 and diagonal is A354595.

Sequence in context: A221877 A110985 A153216 * A295943 A296804 A296952

Adjacent sequences: A327256 A327257 A327258 * A327260 A327261 A327262

Keyword

nonn,tabl

Author

David Lovler, Aug 27 2019

Status

approved