Triangles Exhibiting Ascending Diagonal Properties (Part III)

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 three following triangles in this page. The triangles are shown in Tables I, II and III along with their summed ascending diagonals. These sums, starting at n = 1, correspond, respectively, to the sequence in the OEIS database for the Sloane number A071619 with the formula:

to the sequence for the Sloane number A008577 with the formula:

and to the Sloane number A299259 with the formulas:

and

The left and right ceiling brackets denotes that the the number is rounded off to the next higher number.

The Three Triangles

The three triangles are named for their first row line, a row line originating in the Pascal triangle, i.e, Table I (T121), Table II (T1331) and Table III (T14641). In addition, the triangle with the Sloane number A004737 and the CAT triangle from Part II may also be alternatively renamed T1 and T11, respectfully. Furthermore, T11, T121, T1331 and T14641 can be constructed from T1 by multiplying (x+1)ⁿ (n=1,2,3,4) with a polynomial having the coefficients of the rows in T1. Moreover, we can do this for all n and thus acquire an infinite number of triangles and, therefore, an infinite number of ascending sequences.

The triangles are displayed in irregular triangle array format with two colors corresponding to the ascending diagonals as shown in the partial Tables I-II where the first row - SD represents the Sum of Diagonals:

From the tables above it can be seen that an ascending sequence from the previous triangle may be used to generate a sequence in the next higher triangle by adding together adjacent numbers starting with n=0. For example, adding 0+1, 1+3, 3+6, 6+11, 11+17... to the sequence (from T121) 1,3,6,11,17,... produces the sequence 1,4,9,17,28,... of T1331. And doing the same to the sequence from T1331 generates the sequence from T14641. In fact, addition of adjacent numbers in the the sequence obtained or the CAT triangle in Part I 1,2,4,7,10,14,19,24,30,37... produces the same sequence 1,3,6,11,17,24,33,43,54,67... as T121.

Coded Examples

Since Table I is symmetrical through the central row of the triangle the formulas derived for the coefficients (C) of the left hand side of the triangle used in coding are:

for the center row:

and for the right hand side:

On the other hand, the formulas for coding for C for Tables II and III as shown in the text files uses a number other formulas to generate each row.

Using these formulas the text files (T121-text, T1331-text and (T14641-text with an n value of 15 a triangle can be displayed as an irregular triangle array of rows. The three coded examples are shown in T121-coded, T1331-coded and T14641-coded.

Go back to Part II. Go to homepage.