The Fibonacci Type Sequences Triangle and its Ascending Diagonals (Part A)

The Fibonacci and Lucas numbers form sequences denoted as F_n and L_n form the sequences Fibonacci and Lucas sequences, respectively, in which each number is the sum of the preceding ones. These sequences are shown in Table I along with their position on the table according to the variable n.

Table I
n	−1	0	1	2	3	4	5	6	7	8	9	10	11
L_n	−1	2	1	3	4	7	11	18	29	47	76	123	199
F_n	1	0	1	1	2	3	5	8	13	21	34	55	89

The sequences for the former start at 0 and 1, and for the latter 2 and 1. Although these are the only two sequences in their family, it is possible to increase the number of sequences far beyond these two, where each new sequence is incremented by 1 for the initial number and where the second number is kept at 1. In order to label all these sequences I've taken the liberty to label each sequence as F_N, where N is a Roman numeral such that F_I is a Fibonacci number while F_II is a Lucas number. Fibonacci type numbers higher than N = II are, therefore, the higher sequences starting with 3 as the initial number.

The Fibonacci type and Interleaved Sequence

These sequences can all be constructed within a Pascal type triangle as in Table II where each of the diagonals is a Fibonacci type sequence. For instance, the next to last and the last left diagonals are the Lucas and Fibonacci sequences, respectively. Diagonals to the left of these two correspond to the other Fibonacci type sequences. The ∑ (sum) of rows also corresponds to the Sloane number on the OEIS database A001911.

Table II The Fibonacci Sequences and Sum of Rows
																			∑
									1										1
								2		1									3
							3		1		2								6
						4		1		3		3							11
					5		1		4		4		5						19
				6		1		5		5		7		8					32
			7		1		6		6		9		11		13				53
		8		1		7		7		11		14		18		21			87
	9		1		8		8		13		17		23		29		34		142
10		1		9		9		15		20		28		37		47		55	231

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:

Table III (The Fibonacci type Sequences/Diagonals)
SD	1	2	4	5	8	10	15	18	26	31	44	52	73	86	120	141	196	230	319
1
2	1
3	1	2
4	1	3	3
5	1	4	4	5
6	1	5	5	7	8
7	1	6	6	9	11	13
8	1	7	7	11	14	18	21
9	1	8	8	13	17	23	29	34
10	1	9	9	15	20	28	37	47	55

1	0	1	1	2	3	5	8	13	21		Δ

where the last row shows the difference (Δ) between terms in the column, numbers which are also follow the Fibonacci sequence. Using these (Δ) values the following equations specify the values of each of the terms in the triangle using two of the initial values from Table I where Fibonacci() are the Fibonacci terms:

Fibonacci(−1)= 1; Fibonacci(0) = 0
a(n,n) = Fibonacci(n+1)
a[n][k] = Fibonacci(k−1) + a[n−1][k]

A computer program giving an expanded version (more rows) of this triangle, but using the Fibonacci formula to derive the terms, is shown in Fibonacci expanded along with the sum of diagonals at the top. A copy of the text file is also shown in Fibonacci text. The computer program was adapted from the program I conceived in Dec 2003 for the partitioning of numbers. See the page on partition numbers.

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.

Δ		1		2		1		3		2		5		3		8		5		13		8		21		13		34		21
SD	1		2		4		5		8		10		15		18		26		31		44		52		73		86		120		141

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

S1	1		4		8		15		26		44		73		120		196		319		518		...
S2	2		5		10		18		31		52		86		141		230		374		607		...

where S1 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.

and where S2 is defined by the two linear recurrence equations

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

In addition, the two latter equations from both groups were incorporated into the FibTriangle text in an interleaved manner in order to generate the interleaved sequence in the program which agrees with the sums of the ascending diagonals.

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 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, a₂ = 2 and a₃ = 4. The following program Interleaved Sequence calculates the terms of the sequence with num=72. The Sequence text is also provided.

Alternatively, the two following equations may be used instead:

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 = 1, a₂ = 2, a₃ = 4 and a₄ = 5.

This completes Part A. To go to Part B. Go back to homepage.

																			∑
									1										1
								2		1									3
							3		1		2								6
						4		1		3		3							11
					5		1		4		4		5						19
				6		1		5		5		7		8					32
			7		1		6		6		9		11		13				53
		8		1		7		7		11		14		18		21			87
	9		1		8		8		13		17		23		29		34		142
10		1		9		9		15		20		28		37		47		55	231

SD	1	2	4	5	8	10	15	18	26	31	44	52	73	86	120	141	196	230	319
1
2	1
3	1	2
4	1	3	3
5	1	4	4	5
6	1	5	5	7	8
7	1	6	6	9	11	13
8	1	7	7	11	14	18	21
9	1	8	8	13	17	23	29	34
10	1	9	9	15	20	28	37	47	55

1	0	1	1	2	3	5	8	13	21		Δ

																			∑
									1										1
								2		1									3
							3		1		2								6
						4		1		3		3							11
					5		1		4		4		5						19
				6		1		5		5		7		8					32
			7		1		6		6		9		11		13				53
		8		1		7		7		11		14		18		21			87
	9		1		8		8		13		17		23		29		34		142
10		1		9		9		15		20		28		37		47		55	231

SD	1	2	4	5	8	10	15	18	26	31	44	52	73	86	120	141	196	230	319
1
2	1
3	1	2
4	1	3	3
5	1	4	4	5
6	1	5	5	7	8
7	1	6	6	9	11	13
8	1	7	7	11	14	18	21
9	1	8	8	13	17	23	29	34
10	1	9	9	15	20	28	37	47	55

1	0	1	1	2	3	5	8	13	21		Δ

																			∑
									1										1
								2		1									3
							3		1		2								6
						4		1		3		3							11
					5		1		4		4		5						19
				6		1		5		5		7		8					32
			7		1		6		6		9		11		13				53
		8		1		7		7		11		14		18		21			87
	9		1		8		8		13		17		23		29		34		142
10		1		9		9		15		20		28		37		47		55	231

SD	1	2	4	5	8	10	15	18	26	31	44	52	73	86	120	141	196	230	319
1
2	1
3	1	2
4	1	3	3
5	1	4	4	5
6	1	5	5	7	8
7	1	6	6	9	11	13
8	1	7	7	11	14	18	21
9	1	8	8	13	17	23	29	34
10	1	9	9	15	20	28	37	47	55

1	0	1	1	2	3	5	8	13	21		Δ