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

1,2

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

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.

Example

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

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 *)

Prog

(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))))
(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

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.

Sequence in context: A079893 A324646 A370180 * A113908 A355624 A065369

Adjacent sequences: A286153 A286154 A286155 * A286157 A286158 A286159

Keyword

nonn,tabl

Author

Antti Karttunen, May 04 2017

Status

approved