A262260 Number of triangles formed by the positions of odd numbers in the first n rows of Pascal's triangle, also known as Tartaglia's triangle.
0, 1, 1, 4, 4, 6, 6, 13, 13, 15, 15, 21, 21, 25, 25, 40, 40, 42, 42, 48, 48, 52, 52, 66, 66, 70, 70, 82, 82, 90, 90, 121, 121, 123, 123, 129, 129, 133, 133, 147, 147, 151, 151, 163, 163, 171, 171, 201, 201, 205, 205, 217, 217, 225, 225, 253, 253, 261, 261, 285, 285, 301, 301, 364, 364
Offset: 0
Examples
Taking Pascal's triangle, removing the even terms and replacing each odd term with a dot, will give you this illustration (the circles are connected with lines to show the sub-triangles):
triangle counts
---------------
row new total
=== === =====
0 o 0 0
/ \
1 o---o 1 1
/ \
2 o o 0 1
/ \ / \
3 o---o---o---o 3 4
/ \
4 o o 0 4
/ \ / \
5 o---o o---o 2 6
/ \ / \
6 o o o o 0 6
/ \ / \ / \ / \
7 o---o---o---o---o---o---o---o 7 13
/ \
8 o o 0 13
.
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Formula example:
given a(46) = 171, a(47) is computed as follows:
A = A001316(46) = 16
B = A001316(44) = 8
C = A001316(44) - 1 = 7
D = A001316((47+1-32)/8) - 1 = 1
a(47) = 171 + 16 + 8 + 7 - 1 = 201
.
.
You can find results for a(n), A, B, C and D in the links section for the first 500 rows.
Links
- Riccardo Perego and Emmanuele Villa, Tartaglia's Triangle Odd Distribution (in Italian), 2012.
- Jon E. Schoenfield, Plot of terms through n=2^14.
- Emmanuele Villa, A,B,C,D coefficients for the first 500 rows.
- Emmanuele Villa, C# Program that calculates the first 50 rows.
- Wikipedia, Pascal's Triangle.
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