This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356754 #48 May 26 2023 14:10:04
%S A356754 2,4,6,7,9,11,11,13,15,17,16,18,20,22,24,22,24,26,28,30,32,29,31,33,
%T A356754 35,37,39,41,37,39,41,43,45,47,49,51,46,48,50,52,54,56,58,60,62,56,58,
%U A356754 60,62,64,66,68,70,72,74,67,69,71,73,75,77,79,81,83,85,87
%N A356754 Triangle read by rows: T(n,k) = ((n-1)*(n+2))/2 + 2*k.
%C A356754 The first column of the triangle is the Lazy Caterer's sequence A000124.
%C A356754 Each subsequent column starts with A000124(n) + (2 * (n-1)).
%C A356754 The first downward diagonal is A046691(n).
%C A356754 Columns and downward diagonals of the triangle identify many sequences (possibly shifted) in the database. Examples can be found in crossrefs below.
%C A356754 The sum of the n-th upward diagonal of the triangle is A356288(n).
%F A356754 T(n,k) = ((n-1) * (n+2))/2 + 2*k.
%F A356754 T(n,k+1) = T(n,k) + 2, k < n.
%e A356754 Triangle T(n,k) begins:
%e A356754 n\k 1 2 3 4 5 6 7 8 9 10 11 ...
%e A356754 1: 2
%e A356754 2: 4 6
%e A356754 3: 7 9 11
%e A356754 4: 11 13 15 17
%e A356754 5: 16 18 20 22 24
%e A356754 6: 22 24 26 28 30 32
%e A356754 7: 29 31 33 35 37 39 41
%e A356754 8: 37 39 41 43 45 47 49 51
%e A356754 9: 46 48 50 52 54 56 58 60 62
%e A356754 10: 56 58 60 62 64 66 68 70 72 74
%e A356754 11: 67 69 71 73 75 77 79 81 83 85 87
%e A356754 ...
%t A356754 Table[((n-1)(n+2))/2+2k,{n,20},{k,n}]//Flatten (* _Harvey P. Dale_, May 26 2023 *)
%o A356754 (Python)
%o A356754 def T(n, k): return ((n-1) * (n+2))//2 + 2*k
%o A356754 for n in range(1, 12):
%o A356754 for k in range(1,(n+1)): print(T(n,k), end = ', ')
%o A356754 (Python)
%o A356754 # Indexed as a linear sequence.
%o A356754 def a000124(n): return n*(n+1)//2 + 1
%o A356754 def a(n):
%o A356754 l = m = 0
%o A356754 for k in range(1,n):
%o A356754 lc = a000124(k - 1)
%o A356754 if n >= lc:
%o A356754 l = lc
%o A356754 m = k
%o A356754 else: break
%o A356754 return n + m + (n - l)
%Y A356754 Cf. A000124, A004120, A046691, A051938, A055999, A056000, A155212, A167487, A167499, A167614, A246172, A334563, A356288.
%K A356754 nonn,tabl,easy
%O A356754 1,1
%A A356754 _Torlach Rush_, Aug 25 2022