Lozanić’s triangle (also referred to as
Losanitsch’s triangle) is a number triangle similar to
Pascal’s triangle (cf.
A007318) where three terms out of four are the sum of the two numbers immediately above it, the one out of four exceptions being that (numbering the rows by
and the entries in each row by
) if
is even and
is odd we subtract the
binomial coefficient ( (n / 2) − 1(k − 1) / 2 ) |
.
[1] The difference between Pascal’s triangle and the Losanitsch’s triangle gives the triangle shown in
A034852 (or
A034877 if we omit the border zeros).
It is named after the Serbian chemist Sima Lozanić (Germanized as Losanitsch), who researched it in his investigation into the symmetries exhibited by rows of paraffins, but has since been found to have applications in graph theory and combinatorics.
The entries for
even and
odd (one out of four) are one half of the corresponding entries of
Pascal’s triangle (cf.
A091043,
A091044) They are highlighted
red in the triangle below.
Lozanić’s triangle
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A005418 Row sums:
2 n − C (2 ( ⌊ n / 2⌋ + 1), ⌊ n / 2⌋ − 1) |
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72
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28
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136
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44
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44
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25
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110
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126
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25
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528
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85
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170
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236
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236
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170
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30
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1056
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12
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1
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36
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110
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A005994: Alkane (or paraffin) numbers .
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2080
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13
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1
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7
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42
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A005993: Alkane (or paraffin) numbers .
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4160
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14
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1
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7
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A002620: Positive quarter-squares, positive square and oblong numbers T (n, 2) = ⌊ n / 2⌋ ⌈ n / 2⌉ | .
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8256
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15
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1
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A008619: Each positive integer twice T (m + 1, 1) = [3 + ( − 1) m + 2 m] | .*
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16512
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16
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A000012: The all 1’s sequence .
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32896
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_______________
* With generating function
.
Recurrence equation
When
is even and
is odd, the recurrence equation is:
otherwise, the default recurrence relation is the same as for Pascal’s triangle, i.e. the triangle cell is the sum of the two cells above:
where
is
choose
(
binomial coefficient).
These two rules may be combined into the ‘single’ rule:
Explicit formula
where
( 00 ) = ( 10 ) = ( 11 ) = 1, ( 01 ) = 0, |
so now the extra term is added when
is not even and
is not odd, which is equivalent to the extra term being added by default and then subtracted when
is even and
is odd (thus cancelling it) corresponding to pattern in the recurrence equation.
Lozanić’s triangle rows
The triangle read by rows gives an infinite sequence of finite sequences:
- {{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}, ...}
whose concatenation give A034851:
-
{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, ...}
Lozanić’s triangle row sums
The
th row sum may be computed with the following formula:
which give A005418:
-
{1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, 8256, 16512, 32896, 65792, 131328, 262656, 524800, 1049600, ...}
which has recurrence equations:
- odd,
- even.
Lozanić’s triangle row alternating sums
The
th row alternating sum may be computed with the following formula:
-
which give (not in OEIS, and not A077957 which does not repeat the first 1, 0 pair):
-
{1, 0, 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, 64, 0, 128, 0, 256, 0, 512, 0, 1024, ...}
Lozanić’s triangle columns
The central column yields A032123:
-
{1, 1, 4, 10, 38, 126, 472, 1716, 6470, 24310, 92504, 352716, 1352540, 5200300, 20060016, ...}
whose
th entry (indexed from
) giving
is obtained with the formulae:
- even,
- odd.
Either of the identical next-to-central columns yields A005654:
-
{1, 2, 6, 19, 66, 236, 868, 3235, 12190, 46252, 176484, 676270, 2600612, 10030008, ...}
whose
th entry (indexed from
) giving
is obtained with the formula:
-
The central column entries interleaved with one of the next-to-central columns yields A034872:
-
{1, 1, 1, 2, 4, 6, 10, 19, 38, 66, 126, 236, 472, 868, 1716, 3235, 6470, 12190, 24310, ...}
Lozanić’s triangle diagonals
The diagonals of Lozanić's triangle give:
* = 0 or - 0:
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the all 1's sequence
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A000012: {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
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* = 1 or - 1:
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each positive integer twice
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A008619: {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, ...}
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* = 2 or - 2:
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the quarter-squares
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A002620: {1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, ...}
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* = 3 or - 3:
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the alkane (or paraffin) numbers
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A005993: {1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146, 182, 231, 280, 344, 408, 489, 570, 670, ...}
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* = 4 or - 4:
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the alkane (or paraffin) numbers
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A005994: {1, 3, 9, 19, 38, 66, 110, 170, 255, 365, 511, 693, 924, 1204, 1548, 1956, 2445, ...}
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* = 5 or - 5:
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the alkane (or paraffin) numbers
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A005995: {1, 3, 12, 28, 66, 126, 236, 396, 651, 1001, 1512, 2184, 3108, 4284, 5832, 7752, ...}
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* = 6 or - 6:
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the alkane (or paraffin) numbers
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A018210: {1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520, 4032, 6216, 9324, 13608, 19440, ...}
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* = 7 or - 7:
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the alkane (or paraffin) numbers
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A018211: {1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912, 25236, 38760, ...}
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* = 8 or - 8:
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the alkane (or paraffin) numbers
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A018212: {1, 5, 25, 85, 255, 651, 1519, 3235, 6470, 12190, 21942, 37854, 63090, 101850, ...}
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* = 9 or - 9:
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the alkane (or paraffin) numbers
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A018213: {1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, ...}
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* = 10 or - 10:
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the alkane (or paraffin) numbers
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A018214: {1, 6, 36, 146, 511, 1512, 4032, 9752, 21942, 46252, 92504, 176484, 323554, 572264, ...}
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* ...
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Lozanić’s triangle diagonals (formulae)
The th entry (indexed from ) of the th diagonal (indexed from ) of Lozanić's triangle is given by:[2]
* = 0 or - 0:
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* = 1 or - 1:
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* = 2 or - 2:
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* = 3 or - 3:
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* = 4 or - 4:
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* = 5 or - 5:
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* = 6 or - 6:
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* = 7 or - 7:
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* = 8 or - 8:
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(series representation of generating function gets ever more unwieldy)
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* = 9 or - 9:
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* = 10 or - 10:
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* ...
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Lozanić’s triangle diagonals (recurrence equations)
The entries of the 1st () diagonal are the partial sums of one out of two entries of the 0th () diagonal:
-
The entries of the 2nd () diagonal are the product of successive entries of the 1st () diagonal:
The entries of the even-numbered () diagonals are the partial sums of the previous diagonals:
Lozanić’s triangle diagonals (generating functions)
A generating function for the ()th diagonal is:
and for the ()th diagonal is obtained by dividing that by :
Examples:
The 1st diagonal { 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... } and the 2nd diagonal { 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... } have generating functions:
-
The 3rd diagonal { 1, 2, 6, 10, 19, 28, 44, 60, ... } and the 4th diagonal { 1, 3, 9, 19, 38, 66, 110, 170, ... } have generating functions:
-
The 5th diagonal { 1, 3, 12, 28, 66, 126, 236, ... } and the 6th diagonal { 1, 4, 16, 44, 110, 236, 472, ... } have generating functions:
-
The 7th diagonal { 1, 4, 20, 60, 170, 396, 868, ... } and the 8th diagonal { 1, 5, 25, 85, 255, 651, 1519, 3235, ... } have generating functions:
-
Lozanić’s triangle half sloped diagonals and Fibonacci numbers
Entries of half sloped diagonals rising from the left have constant sum .
Adding the entries on half sloped diagonals, starting with the 1 on an even row, gives:
- A005207
{1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933, ...}
which is obtained by the formula:
-
Adding the entries on half sloped diagonals, starting with the 1 on an odd row, gives:
- A051450
{1, 2, 5, 12, 30, 76, 195, 504, 1309, 3410, 8900, 23256, 60813, 159094, 416325, 1089648, ...}
which is obtained by the formula:
-
where is the th Fibonacci number.
See also
Notes
References
- S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.