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).
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
KEYWORD
AUTHOR
Torlach Rush, Aug 25 2022
STATUS
approved