A176270 Triangle T(n,m) = 1 + m*(m-n) read by rows, 0 <= m <= n.

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, -2, 1, 1, -3, -5, -5, -3, 1, 1, -4, -7, -8, -7, -4, 1, 1, -5, -9, -11, -11, -9, -5, 1, 1, -6, -11, -14, -15, -14, -11, -6, 1, 1, -7, -13, -17, -19, -19, -17, -13, -7, 1, 1, -8, -15, -20, -23, -24, -23, -20, -15, -8, 1

Offset: 0

Author

Roger L. Bagula, Apr 13 2010

Keywords

• sign
• tabl
• easy

Comments

For GCD(-1 - m,-1 - n + m) = 1, smallest number that cannot be written as a*(-1 - m) + b*(-1 - n + m) with a and b in the nonnegative integers. - Thomas Anton, May 22 2019

Examples

			Triangle begins
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,  -3,  -2,   1;
  1,  -3,  -5,  -5,  -3,   1;
  1,  -4,  -7,  -8,  -7,  -4,   1;
  1,  -5,  -9, -11, -11,  -9,  -5,   1;
  1,  -6, -11, -14, -15, -14, -11,  -6,   1;
  1,  -7, -13, -17, -19, -19, -17, -13,  -7,   1;
  1,  -8, -15, -20, -23, -24, -23, -20, -15,  -8,   1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A005586 (row sums), A077028.

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> k*(k-n)+1 ))); # G. C. Greubel, May 30 2019

Magma

[[k*(k-n)+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 30 2019

Maple

A176270 := proc(n,m)
        1+m*(m-n) ;
end proc: # R. J. Mathar, May 03 2013

Mathematica

Table[k*(k-n)+1, {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 30 2019 *)

PARI

{T(n,k) = k*(k-n)+1}; \\ G. C. Greubel, May 30 2019

Sage

[[k*(k-n)+1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 30 2019

Formula

T(n,m) = binomial(n-m+1,2) + binomial(m+1,2) - binomial(n+1,2) + 1 = m^2 - n*m + 1.

T(n,m) = T(n,n-m).

T(n,m) = 2 - A077028(n,m) for 0 <= m <= n. - Werner Schulte, Nov 10 2020

Extensions

Edited by R. J. Mathar, May 03 2013