A356754 Triangle read by rows: T(n,k) = ((n-1)*(n+2))/2 + 2*k.

2, 4, 6, 7, 9, 11, 11, 13, 15, 17, 16, 18, 20, 22, 24, 22, 24, 26, 28, 30, 32, 29, 31, 33, 35, 37, 39, 41, 37, 39, 41, 43, 45, 47, 49, 51, 46, 48, 50, 52, 54, 56, 58, 60, 62, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87

Offset

1,1

Comments

The first column of the triangle is the Lazy Caterer's sequence A000124.

Each subsequent column starts with A000124(n) + (2 * (n-1)).

The first downward diagonal is A046691(n).

Columns and downward diagonals of the triangle identify many sequences (possibly shifted) in the database. Examples can be found in crossrefs below.

The sum of the n-th upward diagonal of the triangle is A356288(n).

Links

Table of n, a(n) for n=1..66.

Formula

T(n,k) = ((n-1) * (n+2))/2 + 2*k.

T(n,k+1) = T(n,k) + 2, k < n.

Example

Triangle T(n,k) begins:
  n\k   1   2   3   4   5   6   7   8   9  10  11  ...
   1:   2
   2:   4   6
   3:   7   9  11
   4:  11  13  15  17
   5:  16  18  20  22  24
   6:  22  24  26  28  30  32
   7:  29  31  33  35  37  39  41
   8:  37  39  41  43  45  47  49  51
   9:  46  48  50  52  54  56  58  60  62
  10:  56  58  60  62  64  66  68  70  72  74
  11:  67  69  71  73  75  77  79  81  83  85  87
  ...

Mathematica

Table[((n-1)(n+2))/2+2k, {n, 20}, {k, n}]//Flatten (* Harvey P. Dale, May 26 2023 *)

Prog

(Python)
def T(n, k): return ((n-1) * (n+2))//2 + 2*k
for n in range(1, 12):
    for k in range(1, (n+1)): print(T(n, k), end = ', ')
(Python)
# Indexed as a linear sequence.
def a000124(n): return n*(n+1)//2 + 1
def a(n):
    l = m = 0
    for k in range(1, n):
        lc = a000124(k - 1)
        if n >= lc:
            l = lc
            m = k
        else: break
    return n + m + (n - l)

Crossrefs

Cf. A000124, A004120, A046691, A051938, A055999, A056000, A155212, A167487, A167499, A167614, A246172, A334563, A356288.

Sequence in context: A184333 A184411 A052056 * A187974 A193715 A320474

Adjacent sequences: A356751 A356752 A356753 * A356755 A356756 A356757

Keyword

nonn,tabl,easy

Author

Torlach Rush, Aug 25 2022

Status

approved