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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 (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_|)].
(3)
Binary plot for Losanitsch's triangle

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

 sum_(k=1)^na_k=2^(n-2)+2^(|_n/2_|-1)
(4)

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


See also

Number Triangle

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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 Triangle

Cite this as:

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

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