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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.

%I A262260 #110 Jun 11 2021 05:14:42
%S A262260 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,
%T A262260 70,70,82,82,90,90,121,121,123,123,129,129,133,133,147,147,151,151,
%U A262260 163,163,171,171,201,201,205,205,217,217,225,225,253,253,261,261,285,285,301,301,364,364
%N 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.
%C A262260 Named Tartaglia's triangle after the Italian mathematician Niccolò Fontana Tartaglia (1500-1577). - _Amiram Eldar_, Jun 11 2021
%H A262260 Riccardo Perego and Emmanuele Villa, <a href="/A262260/a262260.pdf">Tartaglia's Triangle Odd Distribution (in Italian), 2012</a>.
%H A262260 Jon E. Schoenfield, <a href="/A262260/a262260.png">Plot of terms through n=2^14</a>.
%H A262260 Emmanuele Villa, <a href="/A262260/a262260.txt">A,B,C,D coefficients for the first 500 rows</a>.
%H A262260 Emmanuele Villa, <a href="https://www.dropbox.com/s/oo60d32f04rfs6u/tartaglia.cs?dl=0">C# Program that calculates the first 50 rows</a>.
%H A262260 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle">Pascal's Triangle</a>.
%F A262260 Empirical formula:
%F A262260 a(0)=0; a(1)=1; for n>1, a(n) = a(n-1) + A + B + C - D
%F A262260 where
%F A262260 A = A001316(n-1) if n = 2x+1, 0 otherwise
%F A262260 B = A001316(n-3) if n = 4x+1, 0 otherwise
%F A262260 C = B-1 if n = 8x+1, 0 otherwise
%F A262260 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.
%e A262260 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):
%e A262260                                         triangle counts
%e A262260                                         ---------------
%e A262260    row                                     new total
%e A262260    ===                                     === =====
%e A262260     0                  o                    0    0
%e A262260                       / \
%e A262260     1                o---o                  1    1
%e A262260                     /     \
%e A262260     2              o       o                0    1
%e A262260                   / \     / \
%e A262260     3            o---o---o---o              3    4
%e A262260                 /             \
%e A262260     4          o               o            0    4
%e A262260               / \             / \
%e A262260     5        o---o           o---o          2    6
%e A262260             /     \         /     \
%e A262260     6      o       o       o       o        0    6
%e A262260           / \     / \     / \     / \
%e A262260     7    o---o---o---o---o---o---o---o      7   13
%e A262260         /                             \
%e A262260     8  o                               o    0   13
%e A262260 .
%e A262260 .
%e A262260 Formula example:
%e A262260 given a(46) = 171, a(47) is computed as follows:
%e A262260 A = A001316(46) = 16
%e A262260 B = A001316(44) = 8
%e A262260 C = A001316(44) - 1 = 7
%e A262260 D = A001316((47+1-32)/8) - 1 = 1
%e A262260 a(47) = 171 + 16 + 8 + 7 - 1 = 201
%e A262260 .
%e A262260 .
%e A262260 You can find results for a(n), A, B, C and D in the links section for the first 500 rows.
%Y A262260 Cf. A000120, A001316, A001317, A006943, A088512.
%K A262260 nonn,base
%O A262260 0,4
%A A262260 _Emmanuele Villa_, Nov 09 2016