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A361974
(1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.
4
3, 11, 8, 27, 20, 15, 51, 40, 31, 24, 83, 68, 55, 44, 35, 123, 104, 87, 72, 59, 48, 171, 148, 127, 108, 91, 76, 63, 227, 200, 175, 152, 131, 112, 95, 80, 291, 260, 231, 204, 179, 156, 135, 116, 99, 363, 328, 295, 264, 235, 208, 183, 160, 139, 120, 443, 404
OFFSET
1,1
COMMENTS
We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(1,2) is a row-splitting array. The rows and columns of B(1,2) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(1,2) is given by A163255.
FORMULA
B(1,2) = (b(i,j)), where b(i,j) = w(i, 2j-1) + w(i, 2j) for i >= 1, j >= 1, where (w(i,j)) is the natural number array (A000027).
b(i,j) = 2i + (i + 2j - 2)^2.
EXAMPLE
Corner of B(1,2):
3 11 27 51 83 123 171 227
8 20 40 68 104 148 200 260
15 31 55 87 127 175 231 295
24 44 72 108 152 204 264 332
35 59 91 131 179 235 299 371
48 76 112 156 298 268 336 412
(row 1 of A000027) = (1,2,4,7,11,16,22,29,...), so (row 1 of B(1,2)) = (3,11,27,58,...);
(row 2 of A000027) = (3,5,8,12,17,23,30,38,...), so (row 2 of B(1,2)) = (8,20,40,68,...).
MATHEMATICA
zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[h_, k_] := w[h, 2 k - 1] + w[h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361974 sequence*)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361974 array*)
CROSSREFS
Cf. A000027, A163255, A333029, A361975 (array B(2,1)), A361976 (array B(2,2)).
Sequence in context: A126261 A050097 A330566 * A205120 A070613 A316884
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Apr 01 2023
STATUS
approved