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%I #22 Aug 17 2020 07:37:36
%S 2,3,4,4,5,6,5,6,7,8,6,7,8,9,10,7,8,9,10,11,12,8,9,10,11,12,13,14,9,
%T 10,11,12,13,14,15,16,10,11,12,13,14,15,16,17,18,11,12,13,14,15,16,17,
%U 18,19,20,12,13,14,15,16,17,18,19,20,21,22,13,14,15,16,17,18,19,20,21,22
%N Sums of ordinal references for a triangular table read by columns, top to bottom.
%C The ordinal references (i,j) for a triangular table are arranged as follows:
%C (1,1) (2,1) (3,1)
%C ..... (2,2) (3,2)
%C ........... (3,3)
%C The sequence comprises the sum of each reference in each column, read top to bottom. A similar sequence is A003057, which consists of the sums of the ordinal references for an array read by antidiagonals.
%C Subtriangle of triangle in A051162. - _Philippe Deléham_, Mar 26 2013
%C First 9 rows coincide with triangle A248110; T(n,k) = A002260(n,k) + n; T(2*n-1,n) = A016789(n-1). - _Reinhard Zumkeller_, Oct 01 2014
%H Reinhard Zumkeller, <a href="/A108872/b108872.txt">Rows n = 1..125 of triangle, flattened</a>
%H NRICH discussion thread, <a href="https://web.archive.org/web/20151209013049/http://nrich.maths.org/askedNRICH/edited/3225.html">Floor Function Identity</a>. [Via Wayback Machine]
%F a(n) = a(i, j) = i + j
%F a(n) = A002024(n) + A002260(n) = floor(1/2 + sqrt(2n)) + n - (m(m+1)/2) + 1, where m = floor((sqrt(8n+1) - 1) / 2 ). The floor function may be computed directly by using the expression floor(x) = x + (arctan(cot(Pi*x)) / Pi) - 1/2 (equation from nrich.maths.org, see links).
%F sum(k=0..n, T(n,k)) = A005449(n+1). - _Philippe Deléham_, Mar 26 2013
%e a(1) = (1,1) = 1 + 1 = 2
%e a(2) = (2,1) = 2 + 1 = 3
%e a(3) = (2,2) = 2 + 2 = 4
%e a(4) = (3,1) = 3 + 1 = 4, etc.
%e Triangle begins:
%e 2
%e 3, 4
%e 4, 5, 6
%e 5, 6, 7, 8
%e 6, 7, 8, 9, 10
%e 7, 8, 9, 10, 11, 12
%e 8, 9, 10, 11, 12, 13, 14
%e 9, 10, 11, 12, 13, 14, 15, 16
%e ... - _Philippe Deléham_, Mar 26 2013
%t Flatten[ Table[i + j, {j, 1, 12}, {i, 1, j}]] (* _Jean-François Alcover_, Oct 07 2011 *)
%o (Haskell)
%o a108872 n k = a108872_tabl !! (n-1) !! (k-1)
%o a108872_row n = a108872_tabl !! (n-1)
%o a108872_tabl = map (\x -> [x + 1 .. 2 * x]) [1..]
%o -- _Reinhard Zumkeller_, Oct 01 2014
%Y Cf. A003057.
%Y Cf. A002024, A002260.
%Y Cf. A005408, A005843.
%Y Cf. A016789 (central terms), A248110.
%K easy,nonn,tabl
%O 1,1
%A _Andrew S. Plewe_, Jul 13 2005
%E Offset changed by _Reinhard Zumkeller_, Oct 01 2014
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