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

Author

Emmanuele Villa, Nov 09 2016

Keywords

• nonn
• base

Comments

Named Tartaglia's triangle after the Italian mathematician Niccolò Fontana Tartaglia (1500-1577). - Amiram Eldar, Jun 11 2021

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
.
.
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.

Crossrefs

Cf. A000120, A001316, A001317, A006943, A088512.

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.