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Array associated with squares, by antidiagonals.
6

%I #21 Dec 30 2020 03:01:14

%S 1,2,3,5,4,6,10,7,8,11,17,12,9,13,18,26,19,14,15,20,27,37,28,21,16,22,

%T 29,38,50,39,30,23,24,31,40,51,65,52,41,32,25,33,42,53,66,82,67,54,43,

%U 34,35,44,55,68,83,101,84,69,56,45,36,46,57,70,85,102,122,103,86,71,58,47,48,59,72,87,104,123,145,124,105,88,73,60,49,61,74,89,106,125,146,170,147,126,107,90,75,62,63,76,91,108,127,148,171

%N Array associated with squares, by antidiagonals.

%C Every positive integer occurs exactly once; hence, as a sequence, A185725 is a permutation of the positive integers. The square with corners T(0,0)=1 and T(n,n)=n^2 is occupied by the numbers 1,2,...,n^2.

%H G. C. Greubel, <a href="/A185725/b185725.txt">Table of n, a(n) for the first 50 rows, flattened</a>

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

%F T(n,k)=(k-1)^2+2n-1 if n<=k; T(n,k)=(n-1)^2+2k if n>k.

%e Northwest corner:

%e 1...2...5...10...17

%e 3...4...7...12...19

%e 6...8...9...14...21

%e 11..13..15..16...23

%t f[n_,k_]:=(k-1)^2+2*n-1/; n<=k;

%t f[n_,k_]:=(n-1)^2+2*k/; n>k;

%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]

%t Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%Y Cf. A185726, A060734, A185728.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 01 2011