OFFSET
1,4
COMMENTS
As a triangular array read by rows (0; 1, 1; 3, 2, 3; 6, 4, 4, 6; ...), T(n,j) = (1/2)n(n+1-2j)+j(j-1) (1<=j<=n) is the sum of the distances from the vertex j of the path graph 1-2-...-n to all the other vertices. Example: T(4,2)=4 because in the path 1-2-3-4 the distances from vertex 2 to the vertices 1, 2, 3, 4 are 1, 0, 1, 2, respectively; 1+0+1+2=4.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).
FORMULA
R(m,n) = (m(m-1)+n(n-1))/2.
The sum of the terms in the upper left r X r submatrix is Sum_{n=1..r} Sum_{m=1..r} R(n,m) = A112742(r). - J. M. Bergot, Jun 18 2013
EXAMPLE
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...
3, 4, 6, 9, 13, 18, 24, 31, 39, 48, ...
6, 7, 9, 12, 16, 21, 27, 34, 42, 51, ...
10, 11, 13, 16, 20, 25, 31, 38, 46, 55, ...
15, 16, 18, 21, 25, 30, 36, 43, 51, 60, ...
21, 22, 24, 27, 31, 36, 42, 49, 57, 66, ...
28, 29, 31, 34, 38, 43, 49, 56, 64, 73, ...
36, 37, 39, 42, 46, 51, 57, 64, 72, 81, ...
45, 46, 48, 51, 55, 60, 66, 73, 81, 90, ...
R(2,4) = binomial(2,2) + binomial(4,2) = 1 + 6 = 7.
MAPLE
T := proc (n, j) if j <= n then (1/2)*n*(n+1-2*j)+j*(j-1) else 0 end if end proc: for n to 12 do seq(T(n, j), j = 1 .. n) end do; # yields sequence in triangular form
MATHEMATICA
Table[n(n-m-1)+m(m+1)/2, {m, 15}, {n, m}] (* Paolo Xausa, Dec 21 2023 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Sep 14 2008
STATUS
approved