Losanitsch's Triangle -- from Wolfram MathWorld

Number Theory > Special Numbers > Number Triangles >

Losanitsch's Triangle

1
1 1
1 1 1
1 2 2 1
1 2 4 2 1
1 3 6 6 3 1
1 3 9 10 9 3 1
1 4 12 19 19 12 4 1
1 4 16 28 38 28 16 4 1
1 5 20 44 66 66 44 20 5 1
1 5 25 60 110 126 110 60 25 5 1

(1)

Losanitsch's triangle (Sloane's A034851) is a number triangle for which each term is the sum of the two numbers immediately above it, except that, numbering the rows by , 1, 2, ... and the entries in each row by , 1, 2, ..., , are given by the recurrence equations

(2)

where is a binomial coefficient.

can be written in closed form as

(3)

The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Losanitsch's triangle.

The row sums of Losanitsch's triangle are

(4)

the first few terms of which are 1, 2, 3, 6, 10, 20, 36, ... (Sloane's A005418).

REFERENCES:

Losanitsch, S. M. "Die Isometrie-Arten bei den Homologen der Paraffin-Reihe." Chem. Ber. 30, 1917-1926, 1897.

Sloane, N. J. A. http://www.research.att.com/~njas/sequences/classic.html#LOSS.

Sloane, N. J. A. Sequences A005418 and A034851 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Losanitsch's Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LosanitschsTriangle.html