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A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

%I A094727 #55 Jun 19 2025 12:53:24
%S A094727 1,2,3,3,4,5,4,5,6,7,5,6,7,8,9,6,7,8,9,10,11,7,8,9,10,11,12,13,8,9,10,
%T A094727 11,12,13,14,15,9,10,11,12,13,14,15,16,17,10,11,12,13,14,15,16,17,18,
%U A094727 19,11,12,13,14,15,16,17,18,19,20,21,12,13,14,15,16,17,18,19,20,21,22,23
%N A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.
%C A094727 All numbers m occur ceiling(m/2) times, see A004526.
%C A094727 The LCM of the n-th row is A076100. - _Michel Marcus_, Mar 18 2018
%H A094727 Reinhard Zumkeller, <a href="/A094727/b094727.txt">Rows n = 1..150 of triangle, flattened</a>
%H A094727 Bruno Berselli, <a href="/A094727/a094727.jpg">Illustration of the initial terms</a>.
%H A094727 László Németh, <a href="https://www.emis.de/journals/JIS/VOL20/Nemeth/nemeth2.html">On the Binomial Interpolated Triangles</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.
%H A094727 Boris Putievskiy, <a href="https://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A094727 T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.
%F A094727 T(n, n - A005843(k)) = A005843(n-k) for 0 <= k <= n/2.
%F A094727 T(n, n - A005408(k)) = A005408(n-k) for 0 <= k < n/2.
%F A094727 T(A005408(n), n) = A016777(n), n >= 0.
%F A094727 Sum_{k=1..n} T(n, k) = A000326(n) (row sums).
%F A094727 T(n, k) = A002024(n,k) + A002260(n,k) - 1. - _Reinhard Zumkeller_, Apr 27 2006
%F A094727 As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - _Carl R. White_, Jul 30 2009
%F A094727 T(n, k) = n+k-1, n >= k >= 1. - _Vincenzo Librandi_, Nov 23 2009 [corrected by _Klaus Brockhaus_, Nov 23 2009]
%F A094727 T(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - _Reinhard Zumkeller_, Jul 25 2012
%F A094727 From _Boris Putievskiy_, Jan 16 2013: (Start)
%F A094727 a(n) = A002260(n) + A003056(n).
%F A094727 a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
%F A094727 From _G. C. Greubel_, Mar 10 2024: (Start)
%F A094727 T(3*n-3, n) = A016813(n-1).
%F A094727 T(4*n-4, n) = A016861(n-1).
%F A094727 Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).
%F A094727 Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).
%F A094727 Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).
%F A094727 Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)
%e A094727 Triangle begins:
%e A094727   1;
%e A094727   2,  3;
%e A094727   3,  4,  5;
%e A094727   4,  5,  6,  7;
%e A094727   5,  6,  7,  8,  9;
%e A094727   6,  7,  8,  9, 10, 11;
%e A094727   7,  8,  9, 10, 11, 12, 13;
%e A094727   8,  9, 10, 11, 12, 13, 14, 15;
%e A094727   9, 10, 11, 12, 13, 14, 15, 16, 17;
%e A094727   ... - _Philippe Deléham_, Mar 30 2013
%t A094727 Table[n + Range[0, n-1], {n, 12}]//Flatten (* _Michael De Vlieger_, Dec 16 2016 *)
%o A094727 (Magma) z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];
%o A094727 (Haskell)
%o A094727 a094727 n k = n + k
%o A094727 a094727_row n = a094727_tabl !! (n-1)
%o A094727 a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]
%o A094727 -- _Reinhard Zumkeller_, Jul 22 2012
%o A094727 (SageMath) flatten([[n+k for k in range(n)] for n in range(1,16)]) # _G. C. Greubel_, Mar 10 2024
%o A094727 (Python)
%o A094727 from math import isqrt
%o A094727 def A094727(n): return ((a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-a)>>1)+n-1 # _Chai Wah Wu_, Jun 19 2025
%Y A094727 Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.
%Y A094727 Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.
%Y A094727 Cf. A094728, A214604, A214661, A319556.
%Y A094727 Cf. A128076 (rows reversed).
%K A094727 nonn,tabl,easy
%O A094727 1,2
%A A094727 _Reinhard Zumkeller_, May 24 2004