Losanitsch's Triangle
Losanitsch's triangle (OEIS 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 👁 n=0
, 1, 2, ... and the entries in each row by 👁 k=0
, 1, 2, ..., 👁 n
, are given by the recurrence equations
| 👁 a(n,k)={a(n-1,k-1)+a(n-1,k)-(n/2-1; (k-1)/2) for n even and k odd; a(n-1,k-1)+a(n-1,k) otherwise, |
(2)
|
where 👁 (n; k)
is a binomial coefficient.
👁 a(n,k)
can be written in closed form as
![👁 a(n,k)=1/2[(n; k)+(n (mod 2); k (mod 2))(|_1/2n_|; |_1/2k_|)].](https://mathworld.wolfram.com/images/equations/LosanitschsTriangle/NumberedEquation3.svg){kind=link}
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
the first few terms of which are 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).
See also
Number TriangleExplore with Wolfram|Alpha
More things to try:
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."Referenced on Wolfram|Alpha
Losanitsch's TriangleCite this as:
Weisstein, Eric W. "Losanitsch's Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LosanitschsTriangle.html