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Losanitsch,s Triangle

المؤلف:  Losanitsch, S. M

المصدر:  "Die Isometrie-Arten bei den Homologen der Paraffin-Reihe." Chem. Ber. 30

الجزء والصفحة:  ...

7-1-2021

1073

Losanitsch's Triangle

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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 , 1, 2, ... and the entries in each row by , 1, 2, ..., , are given by the recurrence equations

{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, " src="https://mathworld.wolfram.com/images/equations/LosanitschsTriangle/NumberedEquation2.gif" style="height:60px; width:444px" />

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where  is a binomial coefficient.

 can be written in closed form as

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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

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

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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. https://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."