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Triangle read by rows in which row n contains n smallest (> 0) numbers which when incremented by n yield a triangular number.
3

%I #20 Jun 01 2020 04:57:56

%S 2,1,4,3,7,12,2,6,11,17,1,5,10,16,23,4,9,15,22,30,39,3,8,14,21,29,38,

%T 48,2,7,13,20,28,37,47,58,1,6,12,19,27,36,46,57,69,5,11,18,26,35,45,

%U 56,68,81,95,4,10,17,25,34,44,55,67,80,94,109,3,9,16,24,33,43,54,66,79,93,108,124

%N Triangle read by rows in which row n contains n smallest (> 0) numbers which when incremented by n yield a triangular number.

%C Row n contains n terms.

%H Gheorghe Coserea, <a href="/A085930/b085930.txt">Rows n = 1..201, flattened</a>

%F T(n, k) = A000217(k+A003056(n))-n. - _David Wasserman_, Feb 23 2004

%e For n = 4 we have row 4 with 2, 6, 11, 17 since 2 + 4 = 3*4/2, 6 + 4 = 4*5/2, 11 + 4 = 5*6/2, 17 + 4 = 6*7/2.

%e Triangle starts:

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

%e [1] 2;

%e [2] 1, 4;

%e [3] 3, 7, 12;

%e [4] 2, 6, 11, 17;

%e [5] 1, 5, 10, 16, 23;

%e [6] 4, 9, 15, 22, 30, 39;

%e [7] 3, 8, 14, 21, 29, 38 48;

%e [8] 2, 7, 13, 20, 28, 37, 47, 58;

%e [9] 1, 6, 12, 19, 27, 36, 46, 57, 69;

%e [10] 5, 11, 18, 26, 35, 45, 56, 68, 81, 95;

%e [11] 4, 10, 17, 25, 34, 44, 55, 67, 80, 94, 109;

%e [12] ...

%o (PARI)

%o t(n,k) = my(x = (sqrtint(1+8*n)-1)\2); (x+k)*(x+k+1)/2 - n;

%o concat(vector(12, n, vector(n, k, t(n,k)))) \\ _Gheorghe Coserea_, Mar 25 2016

%Y Cf. A000217.

%K easy,nonn,tabl

%O 1,1

%A _Amarnath Murthy_ and _Jason Earls_, Jul 14 2003

%E Offset corrected by _Gheorghe Coserea_, Mar 25 2016