The Ascending Diagonals of a Fibonacci Type Sequences Triangle (Part B)

The triangle in part A is similar to one which can be contructed from the table in the OEIS database A353595 except that a pair of Fibonacci sequences are present in the last two right diagonals of the triangle. We can display this triangle 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:

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 are two interleaved sequences of Fibonacci numbers shifted over 3 terms from each other.

Table I (The Fibonacci type Sequences/Diagonals)
SD013 468 1215 212636 446073 99120162 196264319
0
11
21 1
31 22
41 333
51 445 5
61 55 788
71 669 111313
81 77 111418 2121
91 8813 172329 3434
Δ1212 243 6510816 132621
SD0134 6812 152126364460 7399120

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

S103 61221 366099 162264429...
S214 81526 4473120 196319518...

where S1 is defined by either of the two linear recurrence equations

a(n) = Lucas(n + 2) − 3 − Fibonacci(n)
a(0) = 0, a(1) = 0, a(n) = a(n − 1) + a(n − 2) + 3

and where S2 is defined by either of the two linear recurrence equations

a(n) = Lucas(n + 2) − 3
a(0) = 0, a(1)= 1, a(n) = a(n − 1) + a(n − 2) + 3

which is stored on the OEIS database under the Sloane number A027961.

The Equations for the Interleaved Sequence

Two possible equations may be used one: One where the input number num is even and one where it is odd. Either one gives the same sequence:

for n = 1 or 2   a(n + 1) = a(n − 1) + 3
for even n   a(½(n + 4)) = a(½(n − 4)) + a(½(n) + 3
for odd n   a(½(n + 5)) = a(½(n + 1)) + a(½(n − 3)) + 3

where in the former n is an even number ≥ 4 and in the latter n is an odd number ≥ 3 given that for both equations a0 = 0, a1 = 1, while a2 = 3 and a3 = 4 are derived from the first equation. The following program Interleaved Sequence calculates the terms of the sequence with num=72. The Sequence text is also provided.

Alternatively, the following simpler method may be used instead:

a(2n-1) = a(2n − 2) + F[n]
a(2n) = a(2n − 1) + F[n]

where F is a Fibonacci term used when determining the differences (Δ) between adjacent terms in the sequence SD above. A program using these two equations as an alternative simpler program and its accompanying text are also included.

A third method, shown below, is a third alternative:

for n = 1: a(2n − 1) = a(2n − 2 ) + 3
for n = 2: a(2n) = a(2n − 2 ) + 3
a(2n) = a(2n − 2) + a(2n − 4) + 3
a(2n − 1) = a(2n − 3) + a(2n − 5) + 3

for even and odd values of n > 2 given that a1 = 0, a2 = 1, while a3 = 3 and a4 = 4 are derived from the first two equations.

This completes Part B. Go back to homepage.

Copyright © 2022 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com