login
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.
1
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
0,4
COMMENTS
Named Tartaglia's triangle after the Italian mathematician Niccolò Fontana Tartaglia (1500-1577). - Amiram Eldar, Jun 11 2021
FORMULA
Empirical formula:
a(0)=0; a(1)=1; for n>1, a(n) = a(n-1) + A + B + C - D
where
A = A001316(n-1) if n = 2x+1, 0 otherwise
B = A001316(n-3) if n = 4x+1, 0 otherwise
C = B-1 if n = 8x+1, 0 otherwise
D = A088512(n+1) = A001316((n+1-m)/8)-1 if n = 8x+1, 0 otherwise, where m is the highest power of 2 less than n.
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