OFFSET
0,4
COMMENTS
Named Tartaglia's triangle after the Italian mathematician Niccolò Fontana Tartaglia (1500-1577). - Amiram Eldar, Jun 11 2021
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.
FORMULA
EXAMPLE
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
.
.
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.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Emmanuele Villa, Nov 09 2016
STATUS
approved