A210536 T(n,k) = 3*n + (k-1) mod 3 - 2; n , k > 0, read by antidiagonals.

1, 2, 4, 3, 5, 7, 1, 6, 8, 10, 2, 4, 9, 11, 13, 3, 5, 7, 12, 14, 16, 1, 6, 8, 10, 15, 17, 19, 2, 4, 9, 11, 13, 18, 20, 22, 3, 5, 7, 12, 14, 16, 21, 23, 25, 1, 6, 8, 10, 15, 17, 19, 24, 26, 28, 2, 4, 9, 11, 13, 18, 20, 22, 27, 29, 31, 3, 5, 7, 12, 14, 16, 21, 23, 25, 30, 32, 34

Offset

1,2

Comments

Columns 3*k-2 are A016777,

Columns 3*k-1 are A016789,

Columns 3*k are A008585.

Rows 1 is A010882.

Links

Table of n, a(n) for n=1..78.

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Formula

a(n) = 3*A002260(n) + (A004736(n) - 1) mod 3 - 2. a(n) = 3*i + (j-1) mod 3 - 2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

Example

The start of the sequence as table:
  1....2...3...1...2...3...1...2...3...
  4....5...6...4...5...6...4...5...6...
  7....8...9...7...8...9...7...8...9...
  10..11..12..10..11..12..10..11..12...
  13..14..15..13..14..15..13..14..15...
  16..17..18..16..17..18..16..17..18...
  19..20..21..19..20..21..19..20..21...
  22..23..24..22..23..24..22..23..24...
  25..26..27..25..26..27..25..26..27...
  ...
The start of the sequence as triangle array read by rows:
  1;
  2,4;
  3,5,7;
  1,6,8,10;
  2,4,9,11,13;
  3,5,7,12,14,16;
  1,6,8,10,15,17,19;
  2,4,9,11,13,18,20,22;
  3,5,7,12,14,16,21,23,25;
  ...

Maple

T:= (n, k)-> 3*n+irem(k-1, 3)-2:
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Dec 30 2025

Prog

(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=3*i + (j-1) % 3 - 2

Crossrefs

Cf. A016777, A016789, A008585, A010882, A002260, A004736, A131225.

Sequence in context: A102568 A338834 A338835 * A102767 A356385 A336640

Adjacent sequences: A210533 A210534 A210535 * A210537 A210538 A210539

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 29 2013

Extensions

Row 10 corrected and row 11 completed by Georg Fischer, Dec 30 2025

Status

approved