A309555 Triangle read by rows: T(n,k) = 3 + k*(n-k) for n >= 0, 0 <= k <= n.

3, 3, 3, 3, 4, 3, 3, 5, 5, 3, 3, 6, 7, 6, 3, 3, 7, 9, 9, 7, 3, 3, 8, 11, 12, 11, 8, 3, 3, 9, 13, 15, 15, 13, 9, 3, 3, 10, 15, 18, 19, 18, 15, 10, 3, 3, 11, 17, 21, 23, 23, 21, 17, 11, 3, 3, 12, 19, 24, 27, 28, 27, 24, 19, 12, 3, 3, 13, 21, 27, 31, 33, 33, 31, 27, 21, 13, 3, 3, 14, 23, 30, 35, 38, 39, 38, 35, 30, 23, 14, 3

Offset

0,1

Comments

The rascal triangle (A077028) can be generated by either of the rules South = (East*West+1)/North or South = East+West+1-North; this number triangle can be generated by either of the rules South = (East*West+3)/North or South = East+West+1-North.

It is more suggestive to observe that N*S-E*W = 1 or 3 in the two cases, and (N+S)-(E+W) = 1 in both cases. In fact "3" in the present definition can be replaced by any integer c, and we get a triangle of integers with N*S-E*W = c and (N+S)-(E+W) = 1. I say "suggestive", because these rules also arise in frieze patterns. - N. J. A. Sloane, Aug 28 2019

Links

Table of n, a(n) for n=0..90.

Philip K Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019.

Formula

By rows: a(n,k) = 3 + k(n-k), n >= 0, 0 <= k <= n.

By antidiagonals: T(r,k) = 3 + r*k, r,k >= 0.

Example

For the row n=3: a(3,0)=3, a(3,1)=5, a(3,2)=5, a(3,3)=3, ...
For the antidiagonal r=2: T(2,0)=3, T(2,1)=5, T(2,3)=7, T(2,4)=9, ...
The triangle begins:
..............3..
............3..3..
..........3..4..3..
........3..5...5..3..
......3..6...7...6..3..
....3..7...9...9..7..3..
..3..8..11..12..11..8..3..
3..9..13..15..15..13..9..3.
...

Maple

T:= proc(n, k)
    if n<0 or k<0 or k>n then
       0;
    else
       k*(n-k)+3 ;
    end if;
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

T[n, k]:=k(n-k)+3; T[0, 0] = 3; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Crossrefs

Cf. A077028, A309555, A309557.

Sequence in context: A091799 A276863 A227321 * A262994 A179847 A035936

Adjacent sequences: A309552 A309553 A309554 * A309556 A309557 A309558

Keyword

nonn,tabl

Author

Philip K Hotchkiss, Aug 07 2019

Extensions

Missing a(50)=23 inserted by Georg Fischer, Nov 08 2021

Status

approved