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A108644 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.

%I A108644 #37 Nov 29 2023 17:19:56
%S A108644 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 A108644 27,37,57,44,33,24,20,28,38,50,73,58,45,34,25,29,39,51,65,91,74,59,46,
%U A108644 35,30,40,52,66,82,111,92,75,60,47,36,41,53,67,83,101
%N A108644 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 A108644 The table gives all positive integers exactly once.
%H A108644 G. C. Greubel, <a href="/A108644/b108644.txt">Antidiagonals n = 1..50, flattened</a>
%H A108644 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A108644 From _G. C. Greubel_, Oct 18 2023: (Start)
%F A108644 T(n, k) = A(n-k+1, k) (antidiagonal triangle).
%F A108644 T(n, n) = A002522(n-1).
%F A108644 T(2*n, n) = A005563(n).
%F A108644 T(2*n-1, n) = A000290(n).
%F A108644 T(2*n-2, n) = A002378(n-1), n >= 2.
%F A108644 T(3*n, n) = A033954(n).
%F A108644 Sum_{k=1..n} T(n, k) = A274248(n). (End)
%F A108644 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 A108644 Array begins:
%e A108644    1  2  5 10 17 26 37 ...
%e A108644    3  4  6 11 18 27 38 ...
%e A108644    7  8  9 12 19 28 39 ...
%e A108644   13 14 15 16 20 29 40 ...
%e A108644   21 22 23 24 25 30 41 ...
%e A108644   31 32 33 34 35 36 42 ...
%e A108644   43 44 45 46 47 48 49 ...
%e A108644   ...
%e A108644 Antidiagonal triangle begins as:
%e A108644    1;
%e A108644    3,  2;
%e A108644    7,  4,  5;
%e A108644   13,  8,  6, 10;
%e A108644   21, 14,  9, 11, 17;
%e A108644   31, 22, 15, 12, 18, 26;
%e A108644   43, 32, 23, 16, 19, 27, 37;
%e A108644   ...
%t A108644 A[n_, k_]:= If[k<n, k +n*(n-1), If[k==n, n^2, n +(k-1)^2]];
%t A108644 A108644[n_, k_]:= A[n-k+1,k];
%t A108644 Table[A108644[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 18 2023 *)
%o A108644 (PARI) A(i,j)=if (i==j, i^2, if (i>j, i*(i-1)+j, (j-1)^2+i));
%o A108644 matrix(7,7,n,k,A(n,k)) \\ _Michel Marcus_, Dec 30 2020
%o A108644 (Magma)
%o A108644 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 A108644:= func< n,k | A(n-k+1,k) >;
%o A108644 [A108644(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 18 2023
%o A108644 (SageMath)
%o A108644 def A(n,k):
%o A108644     if k<n: return k+n*(n-1)
%o A108644     elif k==n: return n^2
%o A108644     else: return n+(k-1)^2
%o A108644 def A108644(n,k): return A(n-k+1,k)
%o A108644 flatten([[A108644(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 18 2023
%Y A108644 Cf. A002522 (1st row), A002061 (1st column), A000290 (diagonal).
%Y A108644 Cf. A002378, A002522, A005563, A033954, A098557, A274248.
%K A108644 nonn,tabl
%O A108644 1,2
%A A108644 _Pierre CAMI_, Jun 27 2005