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A322630 Arithmetic table similar to multiplication with separate rules for odd and even products read by antidiagonals. 5
1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 7, 4, 5, 6, 6, 8, 8, 6, 6, 7, 6, 11, 8, 11, 6, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 8, 15, 12, 17, 12, 15, 8, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 10, 19, 16, 23, 18, 23, 16, 19, 10, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.

Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j >0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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

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

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

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

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

EXAMPLE

The array T starts in row n=1 with columns k >= 1 as:

   1   2   3   4   5   6   7   8   9  10

   2   2   4   4   6   6   8   8  10  10

   3   4   7   8  11  12  15  16  19  20

   4   4   8   8  12  12  16  16  20  20

   5   6  11  12  17  18  23  24  29  30

   6   6  12  12  18  18  24  24  30  30

   7   8  15  16  23  24  31  32  39  40

   8   8  16  16  24  24  32  32  40  40

   9  10  19  20  29  30  39  40  49  50

  10  10  20  20  30  30  40  40  50  50

PROG

(PARI) T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));

T(n, k) = (T319929(n, k) + n*k)/2;

matrix(6, 6, n, k, T(n, k)) \\ Michel Marcus, Dec 22 2018

CROSSREFS

Cf. A076274, A327263.

Sequence in context: A074712 A271914 A087735 * A277194 A172151 A106250

Adjacent sequences:  A322627 A322628 A322629 * A322631 A322632 A322633

KEYWORD

nonn,tabl,easy

AUTHOR

David Lovler, Dec 20 2018

STATUS

approved

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Last modified July 8 07:34 EDT 2020. Contains 335513 sequences. (Running on oeis4.)