The Column Addition Triangle (CAT) - A Pascal Δ Analogy (Part I)

Summation of Rising Diagonals in CAT

An interesting property of Pascal's triangle is that its diagonals sum to the Fibonacci sequence as shown online. (Just type in Pascal Fibonacci image). A similar property has been discovered with the column addition triangle (CAT) used to sum up columns in the multiplication of very large numbers. The triangle is shown in Table I along with an image of the same triangle showing ten of its summed ascending diagonals. These sums correspond to the sequence in the OEIS database for the Sloane number A007980 with the formula ⌈n(n+1)/3⌉ starting at n = 1. The left and right ceiling brackets denotes that the the number is rounded off to the next higher number. Table I also contains the sum of the rows which correspond to the Sloane number A001105 with the formula 2n². The CAT triangle is now listed in the OEIS database as Sloane number A347026.

The Triangle and its Image

CAT Image

The CAT Image can also be displayed in irregular triangle array format with two colors corresponding to the ascending diagonals as shown in the partial Table II where the first row - SD represents the Sum of Diagonals:

Since Table I is symmetrical through the center of the triangle the formulas derived for the coefficients (C) of the left hand side of the triangle:

and the right hand side:

are identical and C is an odd number. The difference being, however, that k starts initially at 1 on the left hand side and at n on the right hand side. This differs from the Pascal triangle binomial coefficient formula which has no restrictions on odd/even numbers and k goes in only one direction:

Using this formula (see text file CAT-text) with an n value of 19 the triangle can be displayed as an irregular triangle array which may be read row by row CAT-19.

It will be shown in Part II that the numbers in the CAT triangle are the coefficients of polynomials (as in the Pascal triangle analog) which are divisible by x + 1 producing new polynomials which may or may not be divisible by x + 1 depending on whether the middle coefficient term is odd or even.

In the last table it can be seen that the Sloane number A007980 may also be constructed, accordingly, by adding the red number above it (also a Sloane number A004396) to obtain the next higher number. This red colored sequence is just an even number followed by the same two odds with no accompanying equation.

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