A145917 Triangle read by rows: to get n-th row, start with -4n and successively add 5, 7, 9, 11, 13, ... until reaching a square.

0, -4, 1, -8, -3, 4, -12, -7, 0, 9, -16, -11, -4, 5, 16, -20, -15, -8, 1, 12, 25, -24, -19, -12, -3, 8, 21, 36, -28, -23, -16, -7, 4, 17, 32, 49, -32, -27, -20, -11, 0, 13, 28, 45, 64, -36, -31, -24, -15, -4, 9, 24, 41, 60, 81, -40, -35, -28, -19, -8, 5, 20, 37, 56, 77, 100

Offset: 0

Author

Jared Ricks (jaredricks(AT)yahoo.com), Oct 24 2008

Keywords

• tabl
• sign

Comments

Row n has n+1 entries.

T(n,k) is the discriminant of the quadratic equation x^2+n*x+k=0. - Boris Putievskiy, Jan 11 2013

Examples

			As a triangle:
   0;
  -4,   1;
  -8,  -3,  4;
 -12,  -7,  0, 9;
 -16, -11, -4, 5, 16;
 -20, -15, -8, 1, 12, 25;
  ...
As an array:
  0 -4 -8 -12 -16 -20 -24
  1 -3 -7 -11 -15 -19 -23
  4  0 -4  -8 -12 -16 -20
  9  5  1  -3  -7 -11 -15
 16 12  8   4   0  -4  -8
 25 21 17  13   9   5   1
 36 32 28  24  20  16  12
		

Links

• Boris Putievskiy, Rows n = 1..140 of triangle, flattened
• Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Formula

From Boris Putievskiy, Jan 11 2013: (Start)

T(n,k) = n^2-4*k, n, k >= 0 array read by antidiagonals.

a(n) = (A002260(n)-1)^2 - 4*(A004736(n)-1), n >0.

a(n) = (i-1)^2-4(j-1), n>0, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, and t = floor((-1+sqrt(8*n-7))/2). (End)