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Triangle read by rows: T(n,k) = ((n-1)*(n+2))/2 + 2*k.
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%I #48 May 26 2023 14:10:04

%S 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 35,37,39,41,37,39,41,43,45,47,49,51,46,48,50,52,54,56,58,60,62,56,58,

%U 60,62,64,66,68,70,72,74,67,69,71,73,75,77,79,81,83,85,87

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

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

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

%C The first downward diagonal is A046691(n).

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

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

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

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

%e Triangle T(n,k) begins:

%e n\k 1 2 3 4 5 6 7 8 9 10 11 ...

%e 1: 2

%e 2: 4 6

%e 3: 7 9 11

%e 4: 11 13 15 17

%e 5: 16 18 20 22 24

%e 6: 22 24 26 28 30 32

%e 7: 29 31 33 35 37 39 41

%e 8: 37 39 41 43 45 47 49 51

%e 9: 46 48 50 52 54 56 58 60 62

%e 10: 56 58 60 62 64 66 68 70 72 74

%e 11: 67 69 71 73 75 77 79 81 83 85 87

%e ...

%t Table[((n-1)(n+2))/2+2k,{n,20},{k,n}]//Flatten (* _Harvey P. Dale_, May 26 2023 *)

%o (Python)

%o def T(n, k): return ((n-1) * (n+2))//2 + 2*k

%o for n in range(1, 12):

%o for k in range(1,(n+1)): print(T(n,k), end = ', ')

%o (Python)

%o # Indexed as a linear sequence.

%o def a000124(n): return n*(n+1)//2 + 1

%o def a(n):

%o l = m = 0

%o for k in range(1,n):

%o lc = a000124(k - 1)

%o if n >= lc:

%o l = lc

%o m = k

%o else: break

%o return n + m + (n - l)

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

%K nonn,tabl,easy

%O 1,1

%A _Torlach Rush_, Aug 25 2022