Lozanić's triangle

Lozanić's triangle (sometimes called Losanitsch's triangle) is a geometric arrangement of binomial coefficients in a manner very similar to that of Pascal's triangle. It is named after the Serbian chemist Sima Lozanić, who researched it in his investigation into the symmetries exhibited by rows of paraffins.

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 1listed in OEIS|id=A034851.

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 OEIS2C|id=A004526.

Moving inwards, the next pair of diagonals contain the "quarter-squares" (OEIS2C|id=A002620), or the square numbers and pronic numbers interleaved.

The next pair of diagonals contain the alkane numbers "l"(6, "n") (OEIS2C|id=A005993). And the next pair of diagonals contain the alkane numbers "l"(7, "n") (OEIS2C|id=A005994), while the next pair has the alkane numbers "l"(8, "n") (OEIS2C|id=A005995), then alkane numbers "l"(9, "n") (OEIS2C|id=A018210), then "l"(10, "n") (OEIS2C|id=A018211), "l"(11, "n") (OEIS2C|id=A018212), "l"(12, "n") (OEIS2C|id=A018213), etc.

The sum of the "n"th row of Lozanić's triangle is $2^\{n-2\}\; +\; 2^\{lfloor\; n/2\; floor\; -\; 1\}$ (OEIS2C|id=A005418 lists the first thirty values or so).

The sums of the diagonals of Lozanić's triangle intermix $\{F\_\{2n\; -\; 1\}\; +\; F\_\{n\; +\; 1\; over\; 2$ with $\{F\_\{2n\}\; +\; F\_n\}\; over\; 2$ (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 (OEIS2C|id=A034852, or OEIS2C|id=A034877 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, [http://www.research.att.com/~njas/sequences/classic.html Classic Sequences]