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A286156 Square array read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc.: A(n,k) = T(remainder(n,k), quotient(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table. 13
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 (list; table; graph; refs; listen; history; text; internal format)
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 15x15 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, int(n/k))

for n in xrange(1, 21): print [A(k, n - k + 1) for k in xrange(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: A228549 A079893 A324646 * A113908 A065369 A167772

Adjacent sequences:  A286153 A286154 A286155 * A286157 A286158 A286159

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 04 2017

STATUS

approved

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Last modified August 23 01:10 EDT 2019. Contains 326211 sequences. (Running on oeis4.)