# Triangle A110200 and Variants

Paul D. Hanna pauldhanna at juno.com
Sat Jul 16 17:46:28 CEST 2005

```SeqFans,
Here is a sequence idea that I do not have time to pursue, yet may
be of interest.
Below I describe a triangle of statistics on the BINARY representation of
numbers.
There is an unexpected regularity found in this triangle.

What is the variant of this triangle using TERNARY representation of
numbers?
That is, what is the triangle described by:

Triangle, read by rows, where T(n,k) equals the sum of squares
of numbers < 3^n having exactly k ones in their base 3 expansion.

Many other interesting variants of these triangles exist!
Thanks,
Paul
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Last night I submitted the following sequence A110200:

Triangle, read by rows, where T(n,k) equals the sum of squares
of numbers < 2^n having exactly k ones in their binary expansion.

(I also submitted the triangle of sum of cubes).

Examples.
Row 4 is formed by sums of squares of numbers < 2^4:
T(4,1) = 1^2 + 2^2 + 4^2 + 8^2 = 85;
T(4,2) = 3^2 + 5^2 + 6^2 + 9^2 + 10^2 + 12^2 = 395;
T(4,3) = 7^2 + 11^2 + 13^2 + 14^2 = 535;
T(4,4) = 15^2 = 225.

Formula:
T(n,k) = (4^n-1)/3*C(n-2,k-1) + (2^n-1)^2*C(n-2,k-2)

The triangle begins:
1;
5,9;
21,70,49;
85,395,535,225;
341,1984,3906,3224,961;
1365,9429,24066,29274,17241,3969;
5461,43434,135255,215900,188595,86106,16129; ...
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END

```