POLYGONAL GENERALIZATIONS OF THE FIBONACCI SEQUENCE A. P. AKANDE, MAXWELL SCHNEIDER AND ROBERT SCHNEIDER Abstract. We define a doubly-indexed sequence of “k-Fibonacci” numbers that generalizes the first three cases of classical polygonal numbers, viz. triangular, square and pentagonal numbers, as well as the Fibonacci sequence itself. Similar considerations also yield doubly-indexed generalizations of Lucas sequences.
In commemoration of Fibonacci Day 2020 Recall the first three cases of classical polygonal numbers, viz. triangular numbers , square numbers k 2 , and pentagonal numbers pk = k(3k−1) , which satisfy the tk = k(k+1) 2 2 respective recursions for k ≥ 1: (1)
tk = tk−1 + k,
k 2 = (k − 1)2 + (2k − 1),
pk = pk−1 + (3k + 1).
Surprisingly, these three sequences are special cases of a more general doubly-indexed sequence, that also contains the classical Fibonacci sequence as a case. For n ≥ 0, recall that the nth Fibonacci number Fn is defined by F0 := 0, F1 := 1, and for n ≥ 2 by the recursion (2)
Fn := Fn−1 + Fn−2 ,
e.g. F2 = 1, F3 = 2, F4 = 3, F5 = 5, etc. (see [2, 3]). The recursion (2) is extended to negative indices F−1 = 1, F−2 = −1, F−3 = 2, F−4 = −3, F−5 = 5, etc., using F−n := (−1)n+1 Fn , n ≥ 0. The Fibonacci numbers enjoy the generating function formula (see [1]) ∞ X x (3) = Fn xn , 1 − x − x2 n=0 where the identity is to be understood in terms of formal power series. In this note, we generalize the polygonal numbers above by defining a double sequence Fn,k of k-Fibonacci numbers, a sequence in the n-index of infinite sequences in the k-index. Definition 1. For fixed k ≥ 1, we define the k-Fibonacci numbers Fn,k , n ≥ 0, by F0,k
k X := (j − 1), j=1
F1,k :=
k X (2j − 1), j=1
and for n ≥ 2, the nth k-Fibonacci number is defined recursively: (4)
Fn,k := Fn−1,k + Fn−2,k .
Clearly F0,k = tk−1 , F1,k = k 2 , and F2,k = tk−1 + k 2 = pk , so the sequence Fn,k contains the triangular, square and pentagonal numbers as the n = 0, 1, 2 cases, respectively, as 1
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A. P. AKANDE, MAXWELL SCHNEIDER AND ROBERT SCHNEIDER
well as the classical Fibonacci numbers when k = 1. We note Fn,k can be extended to negative n-indices F−n,k in the same way as F−n , viz. F−n,k := (−1)n+1 Fn,k , n ≥ 0. Carrying through the recursion (4) yields nice formulas for Fn,k , including a closed form expression in terms of classical Fibonacci and triangular numbers. Theorem 2. For n ≥ 0, k ≥ 1, we have that Fn,k =
k X
(Fn+2 j − Fn+1 ) = Fn+2 tk − Fn+1 k,
j=1
where tk is the kth triangular number. Proof. The identity clearly holds for the initial cases F0,k = tk−1 = F2 tk − F1 k and F1,k = k 2 = F3 tk − F2 k; for n ≥ 2, the theorem follows by induction: (5) Fm,k = Fm−1,k + Fm−2,k = (Fm+1 tk − Fm k) + (Fm tk − Fm−1 k) = Fm+2 tk − Fm+1 k. Since the recursions are identical, the interested reader can confirm that many classical Fibonacci identities extend to k-Fibonacci numbers as well. For instance, the generating function for k-Fibonacci numbers generalizes the classical case (3), and highlights further connections to triangular numbers. Theorem 3. We have the generating function identity ∞ X tk x + tk−1 = Fn,k xn . 1 − x − x2 n=0 Remark. The case k = 1 reduces to the classical identity (3). Proof. We proceed similarly to the standard proof of the generating function for Fn . In full generality, for any Lucas sequence an with arbitrary initialPvalues a0 , a1 and an := n an−1 + an−2 for n ≥ 2, define the formal power series A(x) := ∞ n=0 an x . Then a little algebra gives (6)
(1 − x − x2 )A(x) = (a1 − a0 )x + a0 .
In the case an = Fn,k we have a0 = tk−1 , a1 = k 2 ; noting a1 − a0 = k 2 − tk−1 = tk , then dividing through by 1 − x − x2 completes the proof. Remark. One can compare (6) above with (3) to solve for the coefficients (7)
an = a0 Fn−1 + a1 Fn , n ≥ 2.
Setting a0 = tk−1 , a1 = k 2 , then a little algebra re-proves the right side of Theorem 2. As suggested in the proof above, the classical Fibonacci sequence is an instance of a Lucas sequence, which respects the recursion (2), but with different initial values. An analogous doubly-indexed Lucas sequence would set a0,k , a1,k to be arbitrary sequences in the k-index, and then define an,k := an−1,k + an−2,k for the subsequent terms. We would like to generalize the canonical Lucas sequence Ln , which is the case L0 := 2, L1 := 1, and Ln := Ln−1 + Ln−2 for n ≥ 2. However, it is not a priori evident what sequences should be the “right” initial choices for L0,k and L1,k . Now, it is a classical fact that Ln = Fn+1 + Fn−1 . We can use this equivalent expression for Ln to define a
POLYGONAL GENERALIZATIONS OF THE FIBONACCI SEQUENCE
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family of k-Lucas numbers Ln,k with similar behavior to Fn,k , and with the L0,k , L1,k cases determined by k-Fibonacci sequences for n ≥ 0, k ≥ 1: (8)
Ln,k := Fn+1,k + Fn−1,k .
With this definition we have that Ln,1 = Ln , and for n = 0, 1, one computes L0,k = F1,k + F−1,k = 2k 2 ,
(9)
L1,k = F2,k + F0,k = k(2k − 1), noting L1,k is the kth hexagonal number. For n ≥ 2, recursions for Fn+1,k , Fn−1,k give (10)
Ln,k = Ln−1,k + Ln−2,k
as one expects, which follows from (8) together with (4). Then we have comparable identities for Ln,k to those in Theorem 2 above. Theorem 4. For n ≥ 0, k ≥ 1, we have that Ln,k =
k X
(Ln+2 j − Ln+1 ) = Ln+2 tk − Ln+1 k.
j=1
Proof. We note from (8) together with Theorem 2 that Ln,k is equal to (11)
Fn+1,k + Fn−1,k =
k X
(Fn+3 j − Fn+2 ) +
j=1
=
k X
k X
(Fn+1 j − Fn )
j=1 k X [(Fn+3 + Fn+1 )j − (Fn+2 + Fn )] = (Ln+2 j − Ln+1 ),
j=1
j=1
and is also equal to (12)
(Fn+3 tk − Fn+2 k) + (Fn+1 tk − Fn k) = (Fn+3 + Fn+1 ) tk − (Fn+2 + Fn ) k = Ln+2,k tk − Ln+1,k k.
We observe that Ln,k also admits polygonal interpretations for the cases n = 0, 1 (viz. twin squares and hexagonal numbers, respectively), but does not appear to have one for the n = 2 case, in which L2,k = k(5k − 1)/2. In fact, Fn,k does not continue to generalize polygonal numbers either for n ≥ 3. One wonders: do Fn,k or Ln,k represent well-known sequences in the k-aspect, at other values of n? References [1] G. E. Andrews, Number Theory, Courier Corporation, 1994. [2] M. Livio, The golden ratio: The story of phi, the world’s most astonishing number, Broadway Books, 2008. [3] R. Schneider, Fibonacci numbers and the golden ratio, Parabola 52:3 (2016). Department of Mathematics University of Georgia Athens, Georgia 30602, U.S.A. Email address: agbolade.akande@uga.edu
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A. P. AKANDE, MAXWELL SCHNEIDER AND ROBERT SCHNEIDER
Honors Program University of Georgia Athens, Georgia 30602, U.S.A. Email address: maxwell.schneider@uga.edu Department of Mathematics University of Georgia Athens, Georgia 30602, U.S.A. Email address: robert.schneider@uga.edu
