A230135 Triangle read by rows: T(n, k) = 1 if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) else T(n, k) = 0.

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

Offset

0

Comments

The T(n, k) triangle is related to the Kn1p sums of the ‘Races with Ties’ triangle A035317. See A230447 for the Kn1p sums and see A180662 for the definitions of these triangle sums.

The row sums lead to three sequences and they can, quite surprisingly, be linked with Alcuin’s sequence A005044, see the formulas.

Links

Table of n, a(n) for n=0..104.

E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.

Formula

T(n, k) = 1 if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) else T(n, k) = 0.

sum(T(n, k), k=0..n) = A008624(n) = A026922(n+2) = A059169(n+3) = A005044(n+3) - A005044(n-3).

Example

The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
n/k 0   1   2   3   4   5   6   7
------------------------------------------------
0|  1
1|  0,  0
2|  1,  0,  0
3|  0,  0,  1,  0
4|  1,  0,  0,  0,  1
5|  0,  0,  1,  0,  0,  0
6|  1,  0,  0,  0,  1,  0,  0
7|  0,  0,  1,  0,  0,  0,  1,  0

Maple

T := proc(n, k): if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) then return(1) else return(0) fi: end: seq(seq(T(n, k), k=0..n), n=0..13);

Mathematica

Flatten[Table[Which[Mod[k, 4]==2&&Mod[n, 2]==1, 1, Mod[k, 4]==Mod[ n, 2]== 0, 1, True, 0], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, May 15 2021 *)

Crossrefs

Sequence in context: A304362 A368997 A330682 * A359836 A359835 A353331

Adjacent sequences: A230132 A230133 A230134 * A230136 A230137 A230138

Keyword

nonn,easy,tabl

Author

Johannes W. Meijer, Oct 12 2013

Status

approved