#A110 INTEGERS 25 (2025)
A SIMPLE PROOF OF THE FORMULA FOR THE SUM OF
TRIBONACCI AND TETRANACCI NUMBERS SQUARED
Ka ol G yszka
Ins i u e o Ma hema ics, Uni e si y o he Na ional Educa ion Commission,
K akow, Poland
[email p o ec ed]
Recei ed: 8/25/25, Accep ed: 11/11/25, Published: 11/25/25
Abs ac
We o e a e y sho de i a ion o he sum o he squa es o he T ibonacci and he
Te anacci numbe s wi h a bi a y ini ial condi ions. Ou me hod a oids gene a ing
unc ions and ins ead elies on elemen a y ecu ence iden i ies and elescoping
sums.
1. Mo i a ion and P elimina ies
In he li e a u e, he e a e many a icles ela ed o he sum o squa es o he Fi-
bonacci numbe s [1, 3], he T ibonacci numbe s [2, 6], o he Te anacci numbe s
[4, 5, 9]. Howa d and Coope [1] a emp o de i e a sum o squa es o a bi a y
Fibonacci m-s ep numbe s in hei pape , bu hei esul , p esen ed in Theo em
3.3, lea es oom o u he imp o emen .
Schumache [5] ob ains he explici o mula o he squa es o he Te anacci
numbe s, bu his (long) p oo uses induc ion. To p o e he iden i y, Schumache had
o guess he desi ed o mula. P odinge and Selki k [4] use he gene a ing unc ion
app oach, bu hei pape elies on compu e -assis ed calcula ions. Fu he mo e,
Shah [7] ob ains a o mula o he sum o squa ed T ibonacci numbe s, bu he also
uses an induc ion. He also e e s o Zei lin’s o mula [10], which is mo e complica ed.
Soykan [9] inds he sum o squa es and o he ela ed sums h ough a e y long
compu a ion.
In his a icle, we a e conce ned wi h he sequences Fn, Tnand Qn, which a e
he Fibonacci, he T ibonacci, and he Te anacci numbe s, espec i ely, de ined
o a bi a y ini ial condi ions. Thei espec i e ecu ence ela ions a e de ined as
usual; ha is, o example,
Tn+3 =Tn+2 +Tn+1 +Tn, n ≥0.
DOI: 10.5281/zenodo.17711694
INTEGERS: 25 (2025) 2
These sequences can be ex ended o nega i e indices ia he ela ion displayed o
he T ibonacci case, which clea ly ansla es o he emaining cases:
Tn=Tn+3 −Tn+2 −Tn+1, n ≤0.
Mo e in o ma ion abou hese sequences can be ound in [8] (en ies A000045,
A000073, and A000078).
The goal o his no e is o p esen a e y simple and di ec de i a ion o he sums
o squa es o T ibonacci and Te anacci numbe s wi h a bi a y ini ial condi ions.
Ou easoning is a simpli ied e sion o he a gumen p oposed by Jakubczyk [2]
and also applies o sequences wi h a bi a y ini ial condi ions. We also no e ha
ou p oo is sel -con ained.
2. Main Resul s
2.1. Sum o Squa es o he T ibonacci Numbe s
Le us in oduce he no a ion:
S(n) =
n
X
k=1
T2
k, R(n) =
n
X
k=1
TkTk+1, P (n) =
n
X
k=1
TkTk−2.
Lemma 1. The closed- o m exp ession o he sum P(n)is
P(n) = (Tn+1 −Tn−1)2−(T1−T−1)2
4.(1)
P oo . No ice ha
(Tk+1 −Tk−1)2−(Tk−Tk−2)2
= (Tk+1 −Tk−Tk−1+Tk−2)(Tk+1 +Tk−Tk−1−Tk−2)
= 4TkTk−2,
so using he elescoping summa ion we immedia ely ge Equa ion (1).
Theo em 1. The sum S(n)admi s he closed o m
S(n) = 4TnTn+1 −4T0T1−(Tn+1 −Tn−1)2+ (T1−T−1)2
4.(2)
P oo . No e ha
R(n) =
n
X
k=1
(T2
k+TkTk−1+TkTk−2)=S(n) + (T0T1+R(n)−TnTn+1)+P(n).
INTEGERS: 25 (2025) 3
This gi es
S(n) = TnTn+1 −T0T1−P(n)
and he o mula (2) ollows.
Le us no e ha he simple e sion o he jus -p esen ed p oo wo ks o he
Fibonacci numbe s. Namely, we ha e
¯
R(n) :=
n
X
k=1
FkFk+1 =
n
X
k=1
F2
k+
n
X
k=1
FkFk−1=
n
X
k=1
F2
k+ (F0F1+¯
R(n)−FnFn+1)
om which we ob ain n
X
k=1
F2
k=FnFn+1 −F0F1.
2.2. Sum o Squa es o he Te anacci Numbe s
We ex end he idea p esen ed in he p oo o he sum o squa ed T ibonacci numbe s
o he Te anacci case. We begin wi h he ollowing lemma.
Lemma 2. The iden i y
3
n
X
k=1
Qk(Qk−2+Qk−3)=(Qn+1 −Qn−1)2−(Q1−Q−1)2
(3)
−Qn−2(Qn+Qn−3)+Q−2(Q0−Q−3)
holds.
P oo . The iden i y again elies on elescoping summa ion. To ind ou how o apply
i , we pe o m a ew mino algeb aic manipula ions. This s a s om a di e ence
o squa es wi h he same s uc u e as in he T ibonacci case:
(Qk+1 −Qk−1)2−(Qk−Qk−2)2
= (Qk+1 −Qk−Qk−1+Qk−2)(Qk+1 +Qk−Qk−1−Qk−2)
= (2Qk−2+Qk−3)(2Qk+Qk−3)
= 4QkQk−2+ 2Qk−2Qk−3+ 2QkQk−3+Q2
k−3
= 3Qk(Qk−2+Qk−3)+Qk−2(Qk+Qk−3)−Qk−3(Qk−Qk−2−Qk−3)
= 3Qk(Qk−2+Qk−3)+Qk−2(Qk+Qk−3)−Qk−3(Qk−1+Qk−4).
Rea anging gi es
3Qk(Qk−2+Qk−3)=(Qk+1 −Qk−1)2−(Qk−Qk−2)2
−Qk−2(Qk+Qk−3)+Qk−3(Qk−1+Qk−4)
om which he lemma is easily concluded.
INTEGERS: 25 (2025) 4
We a e now eady o de i e he sum o squa es o he Te anacci numbe s. The
mo i a ion o conside ing Equa ion (3) becomes clea when we a emp o epea
he a gumen used o de i ing S(n).
Theo em 2. I holds ha
n
X
k=1
Q2
k=3QnQn+1 −3Q0Q1−(Qn+1 −Qn−1)2+ (Q1−Q−1)2
3
+Qn−2(Qn+Qn−3)−Q−2(Q0−Q−3)
3.
P oo . No e ha
n
X
k=1
QkQk+1 =
n
X
k=1
(Q2
k+QkQk−1+Qk(Qk−2+Qk−3))
=
n
X
k=1
Q2
k+ Q0Q1+
n
X
k=1
QkQk+1 −QnQn+1!+
n
X
k=1
Qk(Qk−2+Qk−3).
Rea anging gi es
n
X
k=1
Q2
k=QnQn+1 −Q0Q1−
n
X
k=1
Qk(Qk−2+Qk−2),
and applying Lemma 2 we ge he inal esul .
Re e ences
[1] C. Coope and F. T. Howa d, Some iden i ies o -Fibonacci numbe s, Fibonacci Qua . 49
(3) (2011), 231–242.
[2] Z. Jakubczyk, Sums o squa es o T ibonacci numbe s (Solu ion o Ad anced P oblem H-715),
Fibonacci Qua . 51 (3) (2013), 285–286.
[3] T. Koshy, Fibonacci and Lucas Numbe s wi h Applica ions, John Wiley and Sons, New Yo k,
2001.
[4] H. P odinge and S. J. Selki k, Sums o squa es o Te anacci numbe s: A gene a ing unc ion
app oach, Fibonacci Qua . 57 (4) (2019), 313–317.
[5] R. Schumache , How o sum he squa es o he Te anacci numbe s and he Fibonacci m-s ep
numbe s, Fibonacci Qua . 57 (2) (2019), 168–175.
[6] R. Schumache , Explici o mulas o sums in ol ing he squa es o he i s nT ibonacci
numbe s, Fibonacci Qua . 58 (3) (2020), 194–202.
[7] D. V. Shah, Some T ibonacci iden i ies, Ma h. Today 27 (2011), 1–9.
[8] N. J. A. Sloane, OEIS Founda ion Inc., The On-Line Encyclopedia o In ege Sequences,
h ps://oeis.o g.
INTEGERS: 25 (2025) 5
[9] Y. Soykan, A s udy on gene alized Te anacci numbe s: Closed o m o mulas Pn
k=0 xkW2
ko
sums o he squa es o e ms, Asian Res. J. Ma h. 16 (10) (2020), 109–136.
[10] D. Zei lin, On summa ion o mulas and iden i ies o Fibonacci numbe s, Fibonacci Qua . 5
(1) (1967), 1–43.
