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A285723 Transpose of square array A285722.

%I A285723 #23 Nov 09 2024 23:45:08
%S A285723 0,1,1,3,0,2,6,2,3,4,10,5,0,5,7,15,9,4,6,8,11,21,14,8,0,9,12,16,28,20,
%T A285723 13,7,10,13,17,22,36,27,19,12,0,14,18,23,29,45,35,26,18,11,15,19,24,
%U A285723 30,37,55,44,34,25,17,0,20,25,31,38,46,66,54,43,33,24,16,21,26,32,39,47,56,78,65,53,42,32,23,0,27,33,40,48,57,67,91,77,64,52,41,31,22,28,34,41,49,58,68,79
%N A285723 Transpose of square array A285722.
%C A285723 See A285722.
%H A285723 Antti Karttunen, <a href="/A285723/b285723.txt">Table of n, a(n) for n = 1..7260; the first 120 antidiagonals of the array</a>
%F A285723 A(n,k) = A285722(k,n).
%e A285723 The top left 14 X 14 corner of the array:
%e A285723    0,  1,  3,  6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91
%e A285723    1,  0,  2,  5,  9, 14, 20, 27, 35, 44, 54, 65, 77, 90
%e A285723    2,  3,  0,  4,  8, 13, 19, 26, 34, 43, 53, 64, 76, 89
%e A285723    4,  5,  6,  0,  7, 12, 18, 25, 33, 42, 52, 63, 75, 88
%e A285723    7,  8,  9, 10,  0, 11, 17, 24, 32, 41, 51, 62, 74, 87
%e A285723   11, 12, 13, 14, 15,  0, 16, 23, 31, 40, 50, 61, 73, 86
%e A285723   16, 17, 18, 19, 20, 21,  0, 22, 30, 39, 49, 60, 72, 85
%e A285723   22, 23, 24, 25, 26, 27, 28,  0, 29, 38, 48, 59, 71, 84
%e A285723   29, 30, 31, 32, 33, 34, 35, 36,  0, 37, 47, 58, 70, 83
%e A285723   37, 38, 39, 40, 41, 42, 43, 44, 45,  0, 46, 57, 69, 82
%e A285723   46, 47, 48, 49, 50, 51, 52, 53, 54, 55,  0, 56, 68, 81
%e A285723   56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,  0, 67, 80
%e A285723   67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,  0, 79
%e A285723   79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,  0
%t A285723 A[n_, n_] = 0;
%t A285723 A[n_, k_] /; k == n - 1 := (k^2 - k + 2)/2;
%t A285723 A[1, k_] := (k^2 - 3 k + 4)/2;
%t A285723 A[n_, k_] /; 1 <= k <= n - 2 := A[n, k] = A[n, k + 1] + 1;
%t A285723 A[n_, k_] /; k > n := A[n, k] = A[n - 1, k] + 1;
%t A285723 T[n_, k_] := A[k, n];
%t A285723 Table[T[n - k + 1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 19 2019 *)
%o A285723 (Scheme) (define (A285723 n) (A285722bi (A004736 n) (A002260 n))) ;; For A285722bi see A285722.
%o A285723 (Python)
%o A285723 def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
%o A285723 def A(n, k): return 0 if n == k else T(n - k, k) if n>k else T(n, k - n)
%o A285723 for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, May 03 2017
%Y A285723 Transpose: A285722.
%Y A285723 Cf. A000217 (row 1), A000124 (column 1, from 1 onward).
%Y A285723 Cf. also A285733.
%K A285723 nonn,tabl
%O A285723 1,4
%A A285723 _Antti Karttunen_, May 03 2017