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Losanitsch's Triangle

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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
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Losanitsch's triangle (OEIS A034851) is a number triangle with rows numbered by n=0, 1, 2, ... and entries in each row numbered by k=0, 1, 2, ..., n. Starting with a(0,0)=1 and taking a(n,k)=0 outside 0<=k<=n, its entries satisfy the recurrence equation

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
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for n>=1, where (n; k) is a binomial coefficient.

For 0<=k<=n, 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_|)],
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where |_x_| is the floor function.

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=0)^na(n,k)=2^(n-1)+2^([n/2]-1)
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for n>=1, where [x] is the ceiling function. Including the row sum 1 for n=0, the first few sums are 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).

See also

Number Triangle

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References

Losanitsch, S. M. "Die Isomerie-Arten bei den Homologen der Paraffin-Reihe." Chem. Ber. 30, 1917-1926, 1897.Sloane, N. J. A. https://oeis.org/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 Resource. https://mathworld.wolfram.com/LosanitschsTriangle.html

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