A286156 A(n,k) = T(remainder(n,k), quotient(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, square array read by descending antidiagonals.

1, 2, 3, 2, 1, 6, 2, 5, 4, 10, 2, 5, 1, 3, 15, 2, 5, 9, 4, 7, 21, 2, 5, 9, 1, 8, 6, 28, 2, 5, 9, 14, 4, 3, 11, 36, 2, 5, 9, 14, 1, 8, 7, 10, 45, 2, 5, 9, 14, 20, 4, 13, 12, 16, 55, 2, 5, 9, 14, 20, 1, 8, 3, 6, 15, 66, 2, 5, 9, 14, 20, 27, 4, 13, 7, 11, 22, 78, 2, 5, 9, 14, 20, 27, 1, 8, 19, 12, 17, 21, 91, 2, 5, 9, 14, 20, 27, 35, 4, 13, 3, 18, 10, 29, 105

Offset: 1

Author

Antti Karttunen, May 04 2017

Keywords

• nonn
• tabl

Examples

			The top left 15 X 15 corner of the array:
    1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,   2,   2
    3,  1,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,   5,   5
    6,  4,  1,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,   9,   9
   10,  3,  4,  1, 14, 14, 14, 14, 14, 14, 14, 14, 14,  14,  14
   15,  7,  8,  4,  1, 20, 20, 20, 20, 20, 20, 20, 20,  20,  20
   21,  6,  3,  8,  4,  1, 27, 27, 27, 27, 27, 27, 27,  27,  27
   28, 11,  7, 13,  8,  4,  1, 35, 35, 35, 35, 35, 35,  35,  35
   36, 10, 12,  3, 13,  8,  4,  1, 44, 44, 44, 44, 44,  44,  44
   45, 16,  6,  7, 19, 13,  8,  4,  1, 54, 54, 54, 54,  54,  54
   55, 15, 11, 12,  3, 19, 13,  8,  4,  1, 65, 65, 65,  65,  65
   66, 22, 17, 18,  7, 26, 19, 13,  8,  4,  1, 77, 77,  77,  77
   78, 21, 10,  6, 12,  3, 26, 19, 13,  8,  4,  1, 90,  90,  90
   91, 29, 16, 11, 18,  7, 34, 26, 19, 13,  8,  4,  1, 104, 104
  105, 28, 23, 17, 25, 12,  3, 34, 26, 19, 13,  8,  4,   1, 119
  120, 37, 15, 24,  6, 18,  7, 43, 34, 26, 19, 13,  8,   4,   1
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
• Eric Weisstein's World of Mathematics, Pairing Function

Crossrefs

Cf. A286157 (transpose), A286158 (lower triangular region), A286159 (lower triangular region transposed).

Cf. A000217 (column 1), A000012 (the main diagonal), A000096 (superdiagonal), A034856.

Cf. A001477, A285722, A286101, A286102.

Programs

Mathematica

Map[((#1 + #2)^2 + 3 #1 + #2)/2 & @@ # & /@ Reverse@ # &, Table[Function[m, Reverse@ QuotientRemainder[m, k]][n - k + 1], {n, 14}, {k, n}]] // Flatten (* Michael De Vlieger, May 20 2017 *)

Python

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n%k, n//k)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)])  # Indranil Ghosh, May 20 2017

Scheme

(define (A286156 n) (A286156bi (A002260 n) (A004736 n)))
(define (A286156bi row col) (if (zero? col) -1 (let ((a (remainder row col)) (b (quotient row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))))

Formula

A(n,k) = T(remainder(n,k), quotient(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...]. This sequence lists only values for indices n >= 1, k >= 1.