A256095 Triangle of greatest common divisors of two triangular numbers (A000217).

0, 1, 1, 3, 1, 3, 6, 1, 3, 6, 10, 1, 1, 2, 10, 15, 1, 3, 3, 5, 15, 21, 1, 3, 3, 1, 3, 21, 28, 1, 1, 2, 2, 1, 7, 28, 36, 1, 3, 6, 2, 3, 3, 4, 36, 45, 1, 3, 3, 5, 15, 3, 1, 9, 45, 55, 1, 1, 1, 5, 5, 1, 1, 1, 5, 55, 66, 1, 3, 6, 2, 3, 3, 2, 6, 3, 11, 66, 78, 1, 3, 6, 2, 3, 3, 2, 6, 3, 1, 6, 78, 91, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 13, 91, 105, 1, 3, 3, 5, 15, 21, 7, 3, 15, 5, 3, 3, 7, 105

Offset: 0

Author

Wolfdieter Lang, Mar 17 2015

Keywords

• nonn
• easy
• tabl

Examples

			The triangle T(n, m) begins:
n\m   0 1 2 3  4  5  6  7  8  9 10 11 12 13  14
0:    0
1:    1 1
2:    3 1 3
3:    6 1 3 6
4:   10 1 1 2 10
5:   15 1 3 3  5 15
6:   21 1 3 3  1  3 21
7:   28 1 1 2  2  1  7 28
8:   36 1 3 6  2  3  3  4 36
9:   45 1 3 3  5 15  3  1  9 45
10:  55 1 1 1  5  5  1  1  1  5 55
11:  66 1 3 6  2  3  3  2  6  3 11 66
12:  78 1 3 6  2  3  3  2  6  3  1  6 78
13:  91 1 1 1  1  1  7  7  1  1  1  1 13 91
14: 105 1 3 3  5 15 21  7  3 15  5  3  3  7 105
...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Crossrefs

Cf. A000217, A144437.

T(2n,n) gives A026741.

Programs

Maple

T:= (i,j) -> igcd(i*(i+1)/2,j*(j+1)/2):
seq(seq(T(i,j),j=0..i),i=0..20); # Robert Israel, Jan 20 2020

PARI

tabl(nn) = {for (n=0, nn, trn = n*(n+1)/2; for (k=0, n, print1(gcd(trn, k*(k+1)/2), ", ");); print(););} \\ Michel Marcus, Mar 17 2015

Formula

T(n, m) = gcd(Tri(n), Tri(m)), 0 <= m <= n, with the triangular numbers Tri = A000217.

T(n, 0) = Tri(n) = T(n, n). T(n, 1) = 1, n >= 0.

Columns m=2: A144437(n-1), m=3: repeat(6, 2, 3, 3, 2, 6, 3, 1, 6, 6, 1, 3) (guess), m=4: repeat(10, 5, 1, 2, 2, 5, 5, 2, 2, 1, 5, 10, 2, 1, 1, 10, 10, 1, 1, 2) (guess), m=5 repeat(15, 3, 1, 3, 15, 5, 3, 3, 1, 15, 15, 1, 3, 3, 5) (guess), ...

From Robert Israel, Jan 21 2020: (Start) The guesses are correct. More generally, for each k>=1, T(n,k) is periodic in n with period 2*A000217(k) if k == 0 or 3 (mod 4), A000217(k) if k == 1 or 2 (mod 4). (End)