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Nonnegative integers in square maze arrangement T(n,k), read by antidiagonals, n>=0, k>=0.
3

%I #33 Jan 02 2021 04:52:32

%S 0,1,3,8,2,4,9,7,5,15,24,10,6,14,16,25,23,11,13,17,35,48,26,22,12,18,

%T 34,36,49,47,27,21,19,33,37,63,80,50,46,28,20,32,38,62,64,81,79,51,45,

%U 29,31,39,61,65,99,120,82,78,52,44,30,40,60,66,98,100

%N Nonnegative integers in square maze arrangement T(n,k), read by antidiagonals, n>=0, k>=0.

%C This sequence consists of 0 together with a permutation of the natural numbers. The structure is the same as A081344 but starting with 0, not 1.

%C It appears that in the n-th layer there is at least a prime number <= g and also there is at least a prime number > g, where g is the number on the main diagonal, the n-th oblong number A002378(n), if n >= 1.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the nonnegative integers</a>

%F a(n) = A081344(n+1) - 1.

%F T(n,k) = n^2 + k , if n is odd and k<=n.

%F T(n,k) = n(n + 2) - k, if n is even and k<=n.

%F T(n,k) = k(k + 2) - n, if n is odd and n<k.

%F T(n,k) = k^2 + n , if n is even and n<k.

%e The first layer is [1, 2, 3] which looks like this:

%e . 3,

%e 1, 2,

%e The second layer is [4, 5, 6, 7, 8] which looks like this:

%e . . 4

%e . . 5,

%e 8, 7, 6,

%e Square array T(0,0)..T(10,10) begins:

%e 0, 3, 4, 15, 16, 35, 36, 63, 64, 99, 100,...

%e 1, 2, 5, 14, 17, 34, 37, 62, 65, 98, 101,...

%e 8, 7, 6, 13, 18, 33, 38, 61, 66, 97, 102,...

%e 9, 10, 11, 12, 19, 32, 39, 60, 67, 96, 103,...

%e 24, 23, 22, 21, 20, 31, 40, 59, 68, 95, 104,...

%e 25, 26, 27, 28, 29, 30, 41, 58, 69, 94, 105,...

%e 48, 47, 46, 45, 44, 43, 42, 57, 70, 93, 106,...

%e 49, 50, 51, 52, 53, 54, 55, 56, 71, 92, 107,...

%e 80, 79, 78, 77, 76, 75, 74, 73, 72, 91, 108,...

%e 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 109,...

%e 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110,...

%e ...

%Y Main diagonal is A002378.

%Y Cf. A000040, A000290, A002620, A081344, A220508.

%K nonn,tabl

%O 0,3

%A _Omar E. Pol_, Feb 09 2013