The Wheel Octagon Triangle - Ascending Diagonals (Part VIIIb)

In the previous page Part VIIIa it was shown that the number of wheel magic squares could be calculated using a slightly different equation for each row. In addition, the table when tilted 45^o on its side resembled a Pascal triangle without the first two rows of ones. Therefore those two rows of ones were added to the truncated Wheel Octagon Triangle (The WOT) to complete the figure. This modified triangle shown below in Table II, though constructed by a series of equations in Part VIIIa, is shown to be constructible below in Table I in an alternative manner. The triangle is known and has a Sloane number in the OEIS database of A162609. What is not known is that this triangle has an ascending diagonal whose sequence is an interleaved sequence.

The terms in each row are obtained by adding a number to each term number to the initial 1 then to each term afterwards until the number of terms equals the row number. For instance, the numbers begin at 0 and are added starting at row 1 (remember row 0 contains the solitary number 1), then incremented and added to the next row and so on. Again, except for the first two rows this triangle is equivalent to Table I of Part VIIIa. Though these three 1s are not included in the table I of Part VIIIa we can extrapolate and substitute n=1 and n=3 in the first equation and n=3 in the second would to make both tables of triangles equivalent.

where the sum (∑) of each row takes on a value from the equation m(m² −3m + 4)/2, starting at m=1, an equation which can be used to derive the sequence stored in the OEIS database under the Sloane number A060354.

Therefore, from the details listed above we can output the WOT using the pseudocode:

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

where the first Print outputs a 1 and the second Print the rest of the terms on the row.

The 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 III where the first row - SD represents the Sum of Diagonals:

A computer program giving an expanded version (more rows) of this triangle is shown in WOT expanded along with the sum of diagonals at the top. A copy of the text file is also shown in WOT text.

The Sum of Diagonal Sequence

The sequence of diagonal terms below is listed along with the delta values, Δ i.e., the difference between terms. The Δ values at first sight do not appear to make sense, but on close inspection and analysis it appears that there appear to be two types of Δ differences here. The blue color ones are the triangular numbers n(n+1)/2 while the known nameless sequence of non blue terms having the equation (3n(n + 1))/2 + 1. Thus, we are dealing with what appears to be the first instance of ascending diagonals whose terms form an interleaved sequence, i.e., SD is composed of two sequences, first one term from one sequence followed by a term from a second sequence. The interleaved sequence is now an entry in the OEIS having a Sloane number A351704.

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

where F₀ = 1 and n ≥ 0.

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

An Alternative Expression for the Interleaved Sequence

Alternatively an expression that can determine each of the terms in the interleaved sequence by calculating the terms as an ordered pair (n₁,n₂) was 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 both equations and what is required is a method that employs two equations, one that calculates the even terms and one that calculates the odd terms of the interleaved equation. To do this the expression above is 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:

This completes Part VIIIb. To go to Part VIIIc to see two new Pascal type triangles in the same family as this one. To go back to Part VIIIa. Go back to homepage.