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Square array A(n,k) read by ascending antidiagonals: A(n,n) = n^2, if n>k: A(n,k) = n*(n-1) + k, if k>n: A(n,k) = n + (k-1)^2.
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%I #37 Nov 29 2023 17:19:56

%S 1,3,2,7,4,5,13,8,6,10,21,14,9,11,17,31,22,15,12,18,26,43,32,23,16,19,

%T 27,37,57,44,33,24,20,28,38,50,73,58,45,34,25,29,39,51,65,91,74,59,46,

%U 35,30,40,52,66,82,111,92,75,60,47,36,41,53,67,83,101

%N Square array A(n,k) read by ascending antidiagonals: A(n,n) = n^2, if n>k: A(n,k) = n*(n-1) + k, if k>n: A(n,k) = n + (k-1)^2.

%C The table gives all positive integers exactly once.

%H G. C. Greubel, <a href="/A108644/b108644.txt">Antidiagonals n = 1..50, flattened</a>

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

%F From _G. C. Greubel_, Oct 18 2023: (Start)

%F T(n, k) = A(n-k+1, k) (antidiagonal triangle).

%F T(n, n) = A002522(n-1).

%F T(2*n, n) = A005563(n).

%F T(2*n-1, n) = A000290(n).

%F T(2*n-2, n) = A002378(n-1), n >= 2.

%F T(3*n, n) = A033954(n).

%F Sum_{k=1..n} T(n, k) = A274248(n). (End)

%F Let M be the upper left n X n submatrix of this array, then abs(det(M)) = A098557(n). - _Thomas Scheuerle_, Nov 11 2023

%e Array begins:

%e 1 2 5 10 17 26 37 ...

%e 3 4 6 11 18 27 38 ...

%e 7 8 9 12 19 28 39 ...

%e 13 14 15 16 20 29 40 ...

%e 21 22 23 24 25 30 41 ...

%e 31 32 33 34 35 36 42 ...

%e 43 44 45 46 47 48 49 ...

%e ...

%e Antidiagonal triangle begins as:

%e 1;

%e 3, 2;

%e 7, 4, 5;

%e 13, 8, 6, 10;

%e 21, 14, 9, 11, 17;

%e 31, 22, 15, 12, 18, 26;

%e 43, 32, 23, 16, 19, 27, 37;

%e ...

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

%t A108644[n_, k_]:= A[n-k+1,k];

%t Table[A108644[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 18 2023 *)

%o (PARI) A(i,j)=if (i==j, i^2, if (i>j, i*(i-1)+j, (j-1)^2+i));

%o matrix(7,7,n,k,A(n,k)) \\ _Michel Marcus_, Dec 30 2020

%o (Magma)

%o A:= func< n,k | k lt n select k+n*(n-1) else k eq n select n^2 else n+(k-1)^2 >;

%o A108644:= func< n,k | A(n-k+1,k) >;

%o [A108644(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 18 2023

%o (SageMath)

%o def A(n,k):

%o if k<n: return k+n*(n-1)

%o elif k==n: return n^2

%o else: return n+(k-1)^2

%o def A108644(n,k): return A(n-k+1,k)

%o flatten([[A108644(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 18 2023

%Y Cf. A002522 (1st row), A002061 (1st column), A000290 (diagonal).

%Y Cf. A002378, A002522, A005563, A033954, A098557, A274248.

%K nonn,tabl

%O 1,2

%A _Pierre CAMI_, Jun 27 2005