The Even/Odd Triangles - Ascending Diagonals (Part VIIIc)

In the previous page Part VIIIb a new Pascal Type Triangle (WOT) was shown along with its ascending diagonals. The terms in the triangle were obtained by adding incremental numbers starting at 0 to each row of the triangle starting at row 1. This section will form two other similar Pascal type triangles (PTTE and PTTO) using either all even numbers or all odd numbers to the triangle rows and show that the sequences obtained are similar in structure to those of the WOT triangle from Part VIIIb. The sequences will also be shown to consists of two interleaved sequences each having their own similar type equations.

The triangles are constructed using the following pseudocode:

                                                              for (n=-1;n<=j;n++){ 
                                                                C=1; 
                                                                Print C;     
                                                                for (k=n; k<=2*n; k++){    
                                                                C=C + N; 
                                                                Print C;     
                                                                 }      
                                                               }

where N equals 2*n for the even derived triangle and 2*n+1 for the odd derived. In addition, the first Print outputs a 1 and the second Print the rest of the terms on the row. For triangle PTTE (Table I) the even numbers were added starting at row 1 (remember row 0 contains the solitary number 1), using the initial even number 0, while for triangle PTTO (Table II) the odd numbers were added starting at again row 1 using the initial odd number 1. The even or odd numbers were then incremented and added subsequently to the next row filling up Tables I and II to the desired number of rows.

The first even triangle is displayed in irregular triangle array format with two colors, blue and yellow, corresponding to the ascending diagonals as shown in the partial Table I where the first row - SD represents the Sum of Diagonals:

where the sum (∑) of each row for PTTE takes may be derived from the equation m³ + 1, starting at m=0, a known sequence stored in the OEIS database under the Sloane number A001093.

Similarly the second odd triangle is displayed as shown in the partial Table II where again the first row - SD represents the Sum of Diagonals:

and where the sum (∑) of each row for PTTO takes may be derived from the equation (2m³ + m² + m + 2)/2, starting at m=0, a known sequence stored in the OEIS database under the Sloane number A081423.

Two computer programs giving expanded versions (more rows) of the PTTE and PTTO triangles are shown in PTTE expanded and PTTO expanded along with the sum of their diagonals at the top. Copies of the text files are also shown in PTTE text and PTTO text.

The Sum of Diagonal Sequence

The first sequence of diagonal terms below is listed along with the delta values, Δ i.e., the difference between terms. There are two types of Δ differences here. The blue color type I (n(n + 1)) and the non blue type II, the hex numbers (3n(n+1) + 1). Again, we are dealing with what ascending diagonals whose terms form an interleaved sequence, i.e., where both SDs are composed of two sequences, first one term from one sequence followed by a term from a second sequence.

The interleaved sequence can be separated into the following two sequences:

where S1 is defined by the linear recurrence equation

where F₀ = 1 and n ≥ 0

and where S2 is defined by the linear recurrence equation

The second sequence of diagonal terms below is listed along with the delta values, Δ i.e., the difference between terms. There are again two types of Δ differences here. The blue color type I (n(n + 1)) as above and the non blue type II, the hexagonal spiral numbers numbers (3n² − 2n + 1), stored in the OEIS database under Sloane number A056105. Again, we are dealing with what ascending diagonals whose terms form an interleaved sequence, i.e., where both SDs are composed of two sequences, first one term from one sequence followed by a term from a second sequence.

The interleaved sequence can be separated into the following two sequences:

where S3 is defined by the linear recurrence equation

where F₀ = 1 and n ≥ 0

and where S4 is defined by the linear recurrence equation

where F₀ = 1 and n ≥ 0.

In addition, these two equations were incorporated into both computer programs in an interleaved manner in order to generate the interleaved sequence in the program which agrees with the sums of the ascending diagonals.

Alternative Expressions for the Interleaved Sequences

Alternatively expressions that can determine each of the terms in the interleaved sequences by calculating the terms as an ordered pair (n₁,n₂) were found, where the left equation (left of the ± sign) is the sum of numbers n1 and n2, while the equation on the right (right of the ± sign) is the difference between n1 and n2:

where n > 0. The right equation in the above expression when subtracted from the left equation and divided by 2 produces n₁ while the right equation when added to the left and divided by 2 produces n₂.

However, this method uses the same n for equation SD1 and the same n for equation SD2 and what is required is a method that employs two equations each, one that calculates the even terms and one that calculates the odd terms of the interleaved equations. To do this the expressions above are separated into two parts (the − and +) and subjected to multiplication of both top and bottom by 6. Replacement of the n of these two equations with the appropriate even [n=(n+2)/2] or odd [n=(n+1)/2] expression, produces the two equations to the right of the arrow. Only even numbers inserted into the first equation and odd into the second equation yield the correct result. For SD1 we have:

and for SD2 we have:

This completes Part VIIIc. To go back to Part VIIIb. Go back to homepage.