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Square array, read by antidiagonals, of the positive integers 6cd +-c +-d = (6c +- 1)d +- c. Alternate rows (or columns) are numbers that differ by c from multiples of 6c - 1 or 6c + 1.
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%I #32 Feb 05 2019 21:03:46

%S 4,6,6,9,8,9,11,13,13,11,14,15,20,15,14,16,20,24,24,20,16,19,22,31,28,

%T 31,22,19,21,27,35,37,37,35,27,21,24,29,42,41,48,41,42,29,24,26,34,46,

%U 50,54,54,50,46,34,26,29,36,53,54,65,60,65,54,53,36,29,31,41,57,63,71,73,73,71,63,57,41,31

%N Square array, read by antidiagonals, of the positive integers 6cd +-c +-d = (6c +- 1)d +- c. Alternate rows (or columns) are numbers that differ by c from multiples of 6c - 1 or 6c + 1.

%C This sequence without duplicates is A067611, which is the complement of A002822, the positive integers x for which 6x - 1 and 6x + 1 are twin primes.

%F a(m,n) = 6*floor((m+1)/2)*floor((n+1)/2) + ((-1)^n)*floor((m+1)/2) + ((-1)^m)*floor((n+1)/2), m,n >= 1.

%e Square array begins:

%e 4, 6, 9, 11, 14, 16, 19, 21, 24, 26, ...

%e 6, 8, 13, 15, 20, 22, 27, 29, 34, 36, ...

%e 9, 13, 20, 24, 31, 35, 42, 46, 53, 57, ...

%e 11, 15, 24, 28, 37, 41, 50, 54, 63, 67, ...

%e 14, 20, 31, 37, 48, 54, 65, 71, 82, 88, ...

%e 16, 22, 35, 41, 54, 60, 73, 79, 92, 98, ...

%e 19, 27, 42, 50, 65, 73, 88, 96, 111, 119, ...

%e 21, 29, 46, 54, 71, 79, 96, 104, 121, 129, ...

%e 24, 34, 53, 63, 82, 92, 111, 121, 140, 150, ...

%e 26, 36, 57, 67, 88, 98, 119, 129, 150, 160, ...

%e ...

%e Note that, for example, the third row (or column) contains numbers that differ by 2 from multiples of 11 = 6*2 - 1, and the eighth row contains numbers that differ by 4 from multiples of 25 = 6*4 + 1.

%o (PARI) a(m,n) = 6*floor((m+1)/2)*floor((n+1)/2) + ((-1)^n)*floor((m+1)/2) + ((-1)^m)*floor((n+1)/2);

%o matrix(7, 7, n, k, a(n, k)) \\ _Michel Marcus_, Jan 25 2019

%Y The first and second rows are A047209 and A047336.

%Y The diagonal is A062717, the numbers x for which 6*x + 1 is a perfect square.

%K nonn,tabl,easy

%O 1,1

%A _Sally Myers Moite_, Jan 23 2019