A085930 Triangle read by rows in which row n contains n smallest (> 0) numbers which when incremented by n yield a triangular number.

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

Offset: 1

Author

Amarnath Murthy and Jason Earls, Jul 14 2003

Keywords

• easy
• nonn
• tabl

Comments

Row n contains n terms.

Examples

			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.
Triangle starts:
n\k  [1]   [2]   [3]   [4]   [5]   [6]   [7]   [8]   [9]   [10]  [11]
[1]  2;
[2]  1,    4;
[3]  3,    7,    12;
[4]  2,    6,    11,   17;
[5]  1,    5,    10,   16,   23;
[6]  4,    9,    15,   22,   30,   39;
[7]  3,    8,    14,   21,   29,   38    48;
[8]  2,    7,    13,   20,   28,   37,   47,   58;
[9]  1,    6,    12,   19,   27,   36,   46,   57,   69;
[10] 5,    11,   18,   26,   35,   45,   56,   68,   81,   95;
[11] 4,    10,   17,   25,   34,   44,   55,   67,   80,   94,   109;
[12] ...
		

Links

• Gheorghe Coserea, Rows n = 1..201, flattened

Crossrefs

Cf. A000217.

Programs

PARI

t(n,k) = my(x = (sqrtint(1+8*n)-1)\2); (x+k)*(x+k+1)/2 - n;
concat(vector(12, n, vector(n, k, t(n,k))))  \\ Gheorghe Coserea, Mar 25 2016

Formula

T(n, k) = A000217(k+A003056(n))-n. - David Wasserman, Feb 23 2004

Extensions

Offset corrected by Gheorghe Coserea, Mar 25 2016