A279416 Triangle read by rows: T(n,k), n >= k >= 1, is the number of grid points below the diagonal of an n X k grid.

0, 0, 1, 0, 2, 3, 0, 3, 5, 6, 0, 4, 6, 9, 10, 0, 5, 8, 11, 14, 15, 0, 6, 9, 12, 16, 20, 21, 0, 7, 11, 15, 19, 23, 27, 28, 0, 8, 12, 17, 20, 26, 30, 35, 36, 0, 9, 14, 18, 24, 29, 33, 39, 44, 45, 0, 10, 15, 21, 26, 30, 37, 43, 48, 54, 55, 0, 11, 17, 23, 29, 35

Offset

1,5

Comments

T(n,1) = 0, T(n,n) = A000217(n-1) = n*(n-1)/2.

Links

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)

Formula

T(n,k) = (k*n - (gcd(n-1, k-1) + 1)) / 2.

Example

Triangle begins:
  0
  0, 1
  0, 2, 3
  0, 3, 5, 6
  0, 4, 6, 9,10
  0, 5, 8,11,14,15
  0, 6, 9,12,16,20,21
  0, 7,11,15,19,23,27,28
  0, 8,12,17,20,26,30,35,36
  0, 9,14,18,24,29,33,39,44,45
  0,10,15,21,26,30,37,43,48,54, 55
  0,11,17,23,29,35,41,47,53,59, 65, 66
  0,12,18,24,30,38,42,51,56,63, 70, 77, 78
  0,13,20,27,34,41,48,55,62,69, 76, 83, 90, 91
  0,14,21,29,36,44,51,56,66,74, 81, 89, 96,104,105
  0,15,23,30,39,45,54,63,71,78, 85, 95,102,111,119,120
  0,16,24,33,40,50,58,67,72,84, 92,101,108,118,126,135,136
  0,17,26,35,44,53,62,71,80,89, 98,107,116,125,134,143,152,153
  0,18,27,36,46,56,63,75,84,90,103,113,120,132,141,150,160,170,171
  0,19,29,39,49,59,69,79,89,99,109,119,129,139,149,159,169,179,189,190

Mathematica

Table[(k n - (GCD[n - 1, k - 1] + 1))/2, {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

Crossrefs

Cf A279417.

Sequence in context: A224909 A227536 A047773 * A331781 A187988 A035549

Adjacent sequences: A279413 A279414 A279415 * A279417 A279418 A279419

Keyword

nonn,tabl,easy

Author

Lars Blomberg, Feb 22 2017

Status

approved