A286234 Square array A(n,k) = P(A000010(k), floor((n+k-1)/k)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

1, 1, 2, 3, 1, 4, 3, 3, 2, 7, 10, 3, 3, 2, 11, 3, 10, 3, 5, 4, 16, 21, 3, 10, 3, 5, 4, 22, 10, 21, 3, 10, 5, 5, 7, 29, 21, 10, 21, 3, 10, 5, 8, 7, 37, 10, 21, 10, 21, 3, 14, 5, 8, 11, 46, 55, 10, 21, 10, 21, 3, 14, 5, 8, 11, 56, 10, 55, 10, 21, 10, 21, 5, 14, 8, 12, 16, 67, 78, 10, 55, 10, 21, 10, 21, 5, 14, 8, 12, 16, 79, 21, 78, 10, 55, 10, 21, 10, 27, 5, 14, 8, 12, 22, 92

Offset

1,3

Comments

Transpose of A286235.

Links

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

MathWorld, Pairing Function

Example

The top left 12 X 12 corner of the array:
   1,  1,  3, 3, 10, 3, 21, 10, 21, 10, 55, 10
   2,  1,  3, 3, 10, 3, 21, 10, 21, 10, 55, 10
   4,  2,  3, 3, 10, 3, 21, 10, 21, 10, 55, 10
   7,  2,  5, 3, 10, 3, 21, 10, 21, 10, 55, 10
  11,  4,  5, 5, 10, 3, 21, 10, 21, 10, 55, 10
  16,  4,  5, 5, 14, 3, 21, 10, 21, 10, 55, 10
  22,  7,  8, 5, 14, 5, 21, 10, 21, 10, 55, 10
  29,  7,  8, 5, 14, 5, 27, 10, 21, 10, 55, 10
  37, 11,  8, 8, 14, 5, 27, 14, 21, 10, 55, 10
  46, 11, 12, 8, 14, 5, 27, 14, 27, 10, 55, 10
  56, 16, 12, 8, 19, 5, 27, 14, 27, 14, 55, 10
  67, 16, 12, 8, 19, 5, 27, 14, 27, 14, 65, 10
The first fifteen rows when viewed as a triangle:
   1
   1  2
   3  1  4
   3  3  2  7
  10  3  3  2 11
   3 10  3  5  4 16
  21  3 10  3  5  4 22
  10 21  3 10  5  5  7 29
  21 10 21  3 10  5  8  7 37
  10 21 10 21  3 14  5  8 11 46
  55 10 21 10 21  3 14  5  8 11 56
  10 55 10 21 10 21  5 14  8 12 16 67
  78 10 55 10 21 10 21  5 14  8 12 16 79
  21 78 10 55 10 21 10 27  5 14  8 12 22 92
  36 21 78 10 55 10 21 10 27  5 19  8 17 22 106

Mathematica

Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ Reverse@ # &, Table[{EulerPhi@ k, Floor[n/k]}, {n, 14}, {k, n}]] // Flatten (* Michael De Vlieger, May 06 2017 *)

Prog

(Scheme)
(define (A286234 n) (A286234bi (A002260 n) (A004736 n)))
(define (A286234bi row col) (let ((a (A000010 col)) (b (quotient (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))
(Python)
from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def t(n, k): return T(totient(k), int(n/k))
for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)][::-1] # Indranil Ghosh, May 11 2017

Crossrefs

Transpose: A286235.

Cf. A000010, A000027, A286156, A286236, A286244.

Sequence in context: A125936 A243614 A200942 * A161621 A095701 A067992

Adjacent sequences: A286231 A286232 A286233 * A286235 A286236 A286237

Keyword

nonn,tabl

Author

Antti Karttunen, May 05 2017

Status

approved