A285733 -id:A285733 - cp's OEIS frontend

A285732 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

-1, 1, 1, 2, -2, 3, 4, 3, 2, 6, 7, 5, -3, 5, 10, 11, 8, 6, 4, 9, 15, 16, 12, 9, -4, 8, 14, 21, 22, 17, 13, 10, 7, 13, 20, 28, 29, 23, 18, 14, -5, 12, 19, 27, 36, 37, 30, 24, 19, 15, 11, 18, 26, 35, 45, 46, 38, 31, 25, 20, -6, 17, 25, 34, 44, 55, 56, 47, 39, 32, 26, 21, 16, 24, 33, 43, 54, 66, 67, 57, 48, 40, 33, 27, -7, 23, 32, 42, 53, 65, 78

Offset: 1

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 14 X 14 corner of the array:
  -1,  1,  2,  4,  7, 11, 16, 22, 29,  37,  46,  56,  67,  79
   1, -2,  3,  5,  8, 12, 17, 23, 30,  38,  47,  57,  68,  80
   3,  2, -3,  6,  9, 13, 18, 24, 31,  39,  48,  58,  69,  81
   6,  5,  4, -4, 10, 14, 19, 25, 32,  40,  49,  59,  70,  82
  10,  9,  8,  7, -5, 15, 20, 26, 33,  41,  50,  60,  71,  83
  15, 14, 13, 12, 11, -6, 21, 27, 34,  42,  51,  61,  72,  84
  21, 20, 19, 18, 17, 16, -7, 28, 35,  43,  52,  62,  73,  85
  28, 27, 26, 25, 24, 23, 22, -8, 36,  44,  53,  63,  74,  86
  36, 35, 34, 33, 32, 31, 30, 29, -9,  45,  54,  64,  75,  87
  45, 44, 43, 42, 41, 40, 39, 38, 37, -10,  55,  65,  76,  88
  55, 54, 53, 52, 51, 50, 49, 48, 47,  46, -11,  66,  77,  89
  66, 65, 64, 63, 62, 61, 60, 59, 58,  57,  56, -12,  78,  90
  78, 77, 76, 75, 74, 73, 72, 71, 70,  69,  68,  67, -13,  91
  91, 90, 89, 88, 87, 86, 85, 84, 83,  82,  81,  80,  79, -14

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array
MathWorld, Pairing Function

Transpose: A285733.
Cf. A000124 (row 1, after -1), A000217 (column 1, after -1).
Cf. also A000027, A003989, A072030, A285721, A285722, A286100.

def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return -n if n == k else T(n - k, k) if n>k else T(n, k - n)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

(define (A285732 n) (A285732bi (A002260 n) (A004736 n)))
(define (A285732bi row col) (cond ((= row col) (- row)) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))

If n = k, A(n,k) = -n, if n > k, A(n,k) = T(n-k,k), otherwise [when n < k], A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N X N to N.

A(n,k) = A285722(n,k) - A286100(n,k).

A285723 Transpose of square array A285722.

Original entry on oeis.org

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, 13, 7, 10, 13, 17, 22, 36, 27, 19, 12, 0, 14, 18, 23, 29, 45, 35, 26, 18, 11, 15, 19, 24, 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

Offset: 1

Antti Karttunen, May 03 2017

See A285722.

			The top left 14 X 14 corner of the array:
   0,  1,  3,  6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91
   1,  0,  2,  5,  9, 14, 20, 27, 35, 44, 54, 65, 77, 90
   2,  3,  0,  4,  8, 13, 19, 26, 34, 43, 53, 64, 76, 89
   4,  5,  6,  0,  7, 12, 18, 25, 33, 42, 52, 63, 75, 88
   7,  8,  9, 10,  0, 11, 17, 24, 32, 41, 51, 62, 74, 87
  11, 12, 13, 14, 15,  0, 16, 23, 31, 40, 50, 61, 73, 86
  16, 17, 18, 19, 20, 21,  0, 22, 30, 39, 49, 60, 72, 85
  22, 23, 24, 25, 26, 27, 28,  0, 29, 38, 48, 59, 71, 84
  29, 30, 31, 32, 33, 34, 35, 36,  0, 37, 47, 58, 70, 83
  37, 38, 39, 40, 41, 42, 43, 44, 45,  0, 46, 57, 69, 82
  46, 47, 48, 49, 50, 51, 52, 53, 54, 55,  0, 56, 68, 81
  56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,  0, 67, 80
  67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,  0, 79
  79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,  0

Antti Karttunen, Table of n, a(n) for n = 1..7260; the first 120 antidiagonals of the array

Transpose: A285722.
Cf. A000217 (row 1), A000124 (column 1, from 1 onward).
Cf. also A285733.

A[n_, n_] = 0;
A[n_, k_] /; k == n - 1 := (k^2 - k + 2)/2;
A[1, k_] := (k^2 - 3 k + 4)/2;
A[n_, k_] /; 1 <= k <= n - 2 := A[n, k] = A[n, k + 1] + 1;
A[n_, k_] /; k > n := A[n, k] = A[n - 1, k] + 1;
T[n_, k_] := A[k, n];
Table[T[n - k + 1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return 0 if n == k else T(n - k, k) if n>k else T(n, k - n)
for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

(define (A285723 n) (A285722bi (A004736 n) (A002260 n))) ;; For A285722bi see A285722.

A(n,k) = A285722(k,n).

cp's OEIS Frontend

A285732 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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