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Rectangular array by antidiagonals: T(n,k) = floor(n + k*r), where r = sqrt(2).
1

%I #10 Jun 01 2022 17:44:43

%S 0,1,1,2,2,2,4,3,3,3,5,5,4,4,4,7,6,6,5,5,5,8,8,7,7,6,6,6,9,9,9,8,8,7,

%T 7,7,11,10,10,10,9,9,8,8,8,12,12,11,11,11,10,10,9,9,9,14,13,13,12,12,

%U 12,11,11,10,10,10,15,15,14,14,13,13,13,12,12,11

%N Rectangular array by antidiagonals: T(n,k) = floor(n + k*r), where r = sqrt(2).

%C Column 0: Nonnegative integers.

%C Row 0: A001951 (Beatty sequence of sqrt(2)).

%C Main diagonal: (0,2,4,7,...): A003151 with 0 prepended.

%F T(n,k) = floor(n + k*r), where r = sqrt(2).

%e Northwest corner:

%e 0 1 2 4 5 7 8 9 11

%e 1 2 3 5 6 8 9 10 12

%e 2 3 4 6 7 9 10 11 13

%e 3 4 5 7 8 10 11 12 14

%e 4 5 6 8 9 11 12 13 15

%e 5 6 7 9 10 12 13 14 16

%t t[n_, k_] := Floor[n + k*Sqrt[2]];

%t Grid[Table[t[n, k], {n, 0, 10}, {k, 0, 10}]] (* A335607 array *)

%t u = Table[t[n - k, k], {n, 0, 13}, {k, n, 0, -1}] // Flatten (* A335607 seq *)

%Y Cf. A001951, A003151, A002193.

%K nonn,tabl,easy

%O 0,4

%A _Clark Kimberling_, Jun 15 2020