Lozanić's triangle (sometimes called Losanitsch's triangle) is a
triangular array of
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s in a manner very similar to that of
Pascal's triangle. It is named after the Serbian chemist
Sima Lozanić
Simeon Milivoje Lozanić and Simeon "Sima" Lozanić ( sr-cyr, Сима Лозанић) (1847 – 1935) was a Serbian chemist, president of the Serbian Royal Academy, the first rector of the University of Belgrade, minister of foreign affairs, mini ...
, who researched it in his investigation into the symmetries exhibited by rows of paraffins (archaic term for
alkane
In organic chemistry, an alkane, or paraffin (a historical trivial name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in which ...
s).
The first few lines of Lozanić's triangle are
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 6 30 85 170 236 236 170 85 30 6 1
1 6 36 110 255 396 472 396 255 110 36 6 1
1 7 42 146 365 651 868 868 651 365 146 42 7 1
1 7 49 182 511 1001 1519 1716 1519 1001 511 182 49 7 1
1 8 56 231 693 1512 2520 3235 3235 2520 1512 693 231 56 8 1
listed in .
Like Pascal's triangle, outer edge diagonals of Lozanić's triangle are all 1s, and most of the enclosed numbers are the sum of the two numbers above. But for numbers at odd positions ''k'' in even-numbered rows ''n'' (starting the numbering for both with 0), after adding the two numbers above, subtract the number at position (''k'' − 1)/2 in row ''n''/2 − 1 of Pascal's triangle.
The diagonals next to the edge diagonals contain the positive integers in order, but with each integer stated twice .
Moving inwards, the next pair of diagonals contain the "quarter-squares" (), or the
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s and
pronic numbers
interleaved.
The next pair of diagonals contain the
alkane numbers ''l''(6, ''n'') (). And the next pair of diagonals contain the alkane numbers ''l''(7, ''n'') (), while the next pair has the alkane numbers ''l''(8, ''n'') (), then alkane numbers ''l''(9, ''n'') (), then ''l''(10, ''n'') (), ''l''(11, ''n'') (), ''l''(12, ''n'') (), etc.
The sum of the ''n''th row of Lozanić's triangle is
( lists the first thirty values or so).
The sums of the diagonals of Lozanić's triangle intermix
with
(where ''F''
''x'' is the ''x''th
Fibonacci number).
As expected, laying Pascal's triangle over Lozanić's triangle and subtracting yields a triangle with the outer diagonals consisting of zeroes (, or for a version without the zeroes). This particular difference triangle has applications in the chemical study of catacondensed polygonal systems.
References
* S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, ''Chem. Ber''. 30 (1897), 1917 - 1926.
* N. J. A. Sloane
Classic Sequences
{{DEFAULTSORT:Lozanic's triangle
Factorial and binomial topics
Triangles of numbers