Products and sum of cubes in Fibonacci
$begingroup$
Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$.
Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a different approach. Hence,
QUESTION. Is there a combinatorial or more conceptual reason for this "pretty" identity?
$$F_nF_{n-1}F_{n-2}=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}3.$$
Caveat. I'm open to as many alternative replies, of course.
Remark. The motivation comes as follows. Define $F_n!=F_1cdots F_n$ and $F_0!=1$. Further, $binom{n}k_F:=frac{F_n!}{F_k!cdot F_{n-k}!}$. Then, I was studying these coefficients and was lead to
$$binom{n}3_F=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}{3!}.$$
nt.number-theory reference-request co.combinatorics elementary-proofs combinatorial-identities
$endgroup$
add a comment |
$begingroup$
Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$.
Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a different approach. Hence,
QUESTION. Is there a combinatorial or more conceptual reason for this "pretty" identity?
$$F_nF_{n-1}F_{n-2}=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}3.$$
Caveat. I'm open to as many alternative replies, of course.
Remark. The motivation comes as follows. Define $F_n!=F_1cdots F_n$ and $F_0!=1$. Further, $binom{n}k_F:=frac{F_n!}{F_k!cdot F_{n-k}!}$. Then, I was studying these coefficients and was lead to
$$binom{n}3_F=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}{3!}.$$
nt.number-theory reference-request co.combinatorics elementary-proofs combinatorial-identities
$endgroup$
$begingroup$
I get a different left hand side. Rewrite the n+2 term as the sum of n+1 and n terms, and then compute the difference of cubes and divide by three. Algebraically you get the product of the n term and the n+1 term and (the sum of n+1 and n terms). This seems to have more to do with (a+b)^n - a^n - b^n than with Fibonacci. Gerhard "Unsure Of Any Combinatorial Interpretation" Paseman, 2019.03.26.
$endgroup$
– Gerhard Paseman
16 hours ago
$begingroup$
Thanks, edited accordingly.
$endgroup$
– T. Amdeberhan
16 hours ago
5
$begingroup$
$(a+b)^3 - a^3 - b^3 = 3ab(a+b)$. If $a,b$ are consecutive Fibonacci numbers then $a+b$ is the next.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
$begingroup$
Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$.
Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a different approach. Hence,
QUESTION. Is there a combinatorial or more conceptual reason for this "pretty" identity?
$$F_nF_{n-1}F_{n-2}=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}3.$$
Caveat. I'm open to as many alternative replies, of course.
Remark. The motivation comes as follows. Define $F_n!=F_1cdots F_n$ and $F_0!=1$. Further, $binom{n}k_F:=frac{F_n!}{F_k!cdot F_{n-k}!}$. Then, I was studying these coefficients and was lead to
$$binom{n}3_F=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}{3!}.$$
nt.number-theory reference-request co.combinatorics elementary-proofs combinatorial-identities
$endgroup$
Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$.
Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a different approach. Hence,
QUESTION. Is there a combinatorial or more conceptual reason for this "pretty" identity?
$$F_nF_{n-1}F_{n-2}=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}3.$$
Caveat. I'm open to as many alternative replies, of course.
Remark. The motivation comes as follows. Define $F_n!=F_1cdots F_n$ and $F_0!=1$. Further, $binom{n}k_F:=frac{F_n!}{F_k!cdot F_{n-k}!}$. Then, I was studying these coefficients and was lead to
$$binom{n}3_F=frac{F_n^3-F_{n-1}^3-F_{n-2}^3}{3!}.$$
nt.number-theory reference-request co.combinatorics elementary-proofs combinatorial-identities
nt.number-theory reference-request co.combinatorics elementary-proofs combinatorial-identities
edited 16 hours ago
T. Amdeberhan
asked 17 hours ago
T. AmdeberhanT. Amdeberhan
18k229131
18k229131
$begingroup$
I get a different left hand side. Rewrite the n+2 term as the sum of n+1 and n terms, and then compute the difference of cubes and divide by three. Algebraically you get the product of the n term and the n+1 term and (the sum of n+1 and n terms). This seems to have more to do with (a+b)^n - a^n - b^n than with Fibonacci. Gerhard "Unsure Of Any Combinatorial Interpretation" Paseman, 2019.03.26.
$endgroup$
– Gerhard Paseman
16 hours ago
$begingroup$
Thanks, edited accordingly.
$endgroup$
– T. Amdeberhan
16 hours ago
5
$begingroup$
$(a+b)^3 - a^3 - b^3 = 3ab(a+b)$. If $a,b$ are consecutive Fibonacci numbers then $a+b$ is the next.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
$begingroup$
I get a different left hand side. Rewrite the n+2 term as the sum of n+1 and n terms, and then compute the difference of cubes and divide by three. Algebraically you get the product of the n term and the n+1 term and (the sum of n+1 and n terms). This seems to have more to do with (a+b)^n - a^n - b^n than with Fibonacci. Gerhard "Unsure Of Any Combinatorial Interpretation" Paseman, 2019.03.26.
$endgroup$
– Gerhard Paseman
16 hours ago
$begingroup$
Thanks, edited accordingly.
$endgroup$
– T. Amdeberhan
16 hours ago
5
$begingroup$
$(a+b)^3 - a^3 - b^3 = 3ab(a+b)$. If $a,b$ are consecutive Fibonacci numbers then $a+b$ is the next.
$endgroup$
– Noam D. Elkies
15 hours ago
$begingroup$
I get a different left hand side. Rewrite the n+2 term as the sum of n+1 and n terms, and then compute the difference of cubes and divide by three. Algebraically you get the product of the n term and the n+1 term and (the sum of n+1 and n terms). This seems to have more to do with (a+b)^n - a^n - b^n than with Fibonacci. Gerhard "Unsure Of Any Combinatorial Interpretation" Paseman, 2019.03.26.
$endgroup$
– Gerhard Paseman
16 hours ago
$begingroup$
I get a different left hand side. Rewrite the n+2 term as the sum of n+1 and n terms, and then compute the difference of cubes and divide by three. Algebraically you get the product of the n term and the n+1 term and (the sum of n+1 and n terms). This seems to have more to do with (a+b)^n - a^n - b^n than with Fibonacci. Gerhard "Unsure Of Any Combinatorial Interpretation" Paseman, 2019.03.26.
$endgroup$
– Gerhard Paseman
16 hours ago
$begingroup$
Thanks, edited accordingly.
$endgroup$
– T. Amdeberhan
16 hours ago
$begingroup$
Thanks, edited accordingly.
$endgroup$
– T. Amdeberhan
16 hours ago
5
5
$begingroup$
$(a+b)^3 - a^3 - b^3 = 3ab(a+b)$. If $a,b$ are consecutive Fibonacci numbers then $a+b$ is the next.
$endgroup$
– Noam D. Elkies
15 hours ago
$begingroup$
$(a+b)^3 - a^3 - b^3 = 3ab(a+b)$. If $a,b$ are consecutive Fibonacci numbers then $a+b$ is the next.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$F_n$ is the number of compositions (ordered partitions) of $n-1$ into
parts equal to 1 or 2. The number of triples $(a,b,c)$ of such
compositions is $F_n^3$. The number such that $a,b,c$ all begin with 1
is $F_{n-1}^3$. The number such that $a,b,c$ all begin with 2 is
$F_{n-2}^3$. Otherwise either one of $a,b,c$ begins with 1 and the
others begin with 2, or vice versa. There are $3F_{n-1}F_{n-2}^2$ such
triples of the first type. Similarly there are $3F_{n-1}^2F_{n-2}$
of the second type, i.e., one of
$a,b,c$ begins with 2 and the others begin with 1. Hence
begin{eqnarray*} F_n^3 & = & F_{n-1}^3 + F_{n-2}^3
+3(F_{n-1}^2F_{n-2}+F_{n-1}F_{n-2}^2)\ & = &
F_{n-1}^3 + F_{n-2}^3 +3F_{n-1}F_{n-2}(F_{n-1}+F_{n-2})\ & = &
F_{n-1}^3 + F_{n-2}^3 + F_nF_{n-1}F_{n-2}. end{eqnarray*}
$endgroup$
2
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
3
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
6
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
add a comment |
$begingroup$
This is just the following identity:
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$ Since $$F_n+(-F_{n-1})+(-F_{n-2})=0,$$ your formula follows.
$endgroup$
7
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$F_n$ is the number of compositions (ordered partitions) of $n-1$ into
parts equal to 1 or 2. The number of triples $(a,b,c)$ of such
compositions is $F_n^3$. The number such that $a,b,c$ all begin with 1
is $F_{n-1}^3$. The number such that $a,b,c$ all begin with 2 is
$F_{n-2}^3$. Otherwise either one of $a,b,c$ begins with 1 and the
others begin with 2, or vice versa. There are $3F_{n-1}F_{n-2}^2$ such
triples of the first type. Similarly there are $3F_{n-1}^2F_{n-2}$
of the second type, i.e., one of
$a,b,c$ begins with 2 and the others begin with 1. Hence
begin{eqnarray*} F_n^3 & = & F_{n-1}^3 + F_{n-2}^3
+3(F_{n-1}^2F_{n-2}+F_{n-1}F_{n-2}^2)\ & = &
F_{n-1}^3 + F_{n-2}^3 +3F_{n-1}F_{n-2}(F_{n-1}+F_{n-2})\ & = &
F_{n-1}^3 + F_{n-2}^3 + F_nF_{n-1}F_{n-2}. end{eqnarray*}
$endgroup$
2
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
3
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
6
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
add a comment |
$begingroup$
$F_n$ is the number of compositions (ordered partitions) of $n-1$ into
parts equal to 1 or 2. The number of triples $(a,b,c)$ of such
compositions is $F_n^3$. The number such that $a,b,c$ all begin with 1
is $F_{n-1}^3$. The number such that $a,b,c$ all begin with 2 is
$F_{n-2}^3$. Otherwise either one of $a,b,c$ begins with 1 and the
others begin with 2, or vice versa. There are $3F_{n-1}F_{n-2}^2$ such
triples of the first type. Similarly there are $3F_{n-1}^2F_{n-2}$
of the second type, i.e., one of
$a,b,c$ begins with 2 and the others begin with 1. Hence
begin{eqnarray*} F_n^3 & = & F_{n-1}^3 + F_{n-2}^3
+3(F_{n-1}^2F_{n-2}+F_{n-1}F_{n-2}^2)\ & = &
F_{n-1}^3 + F_{n-2}^3 +3F_{n-1}F_{n-2}(F_{n-1}+F_{n-2})\ & = &
F_{n-1}^3 + F_{n-2}^3 + F_nF_{n-1}F_{n-2}. end{eqnarray*}
$endgroup$
2
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
3
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
6
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
add a comment |
$begingroup$
$F_n$ is the number of compositions (ordered partitions) of $n-1$ into
parts equal to 1 or 2. The number of triples $(a,b,c)$ of such
compositions is $F_n^3$. The number such that $a,b,c$ all begin with 1
is $F_{n-1}^3$. The number such that $a,b,c$ all begin with 2 is
$F_{n-2}^3$. Otherwise either one of $a,b,c$ begins with 1 and the
others begin with 2, or vice versa. There are $3F_{n-1}F_{n-2}^2$ such
triples of the first type. Similarly there are $3F_{n-1}^2F_{n-2}$
of the second type, i.e., one of
$a,b,c$ begins with 2 and the others begin with 1. Hence
begin{eqnarray*} F_n^3 & = & F_{n-1}^3 + F_{n-2}^3
+3(F_{n-1}^2F_{n-2}+F_{n-1}F_{n-2}^2)\ & = &
F_{n-1}^3 + F_{n-2}^3 +3F_{n-1}F_{n-2}(F_{n-1}+F_{n-2})\ & = &
F_{n-1}^3 + F_{n-2}^3 + F_nF_{n-1}F_{n-2}. end{eqnarray*}
$endgroup$
$F_n$ is the number of compositions (ordered partitions) of $n-1$ into
parts equal to 1 or 2. The number of triples $(a,b,c)$ of such
compositions is $F_n^3$. The number such that $a,b,c$ all begin with 1
is $F_{n-1}^3$. The number such that $a,b,c$ all begin with 2 is
$F_{n-2}^3$. Otherwise either one of $a,b,c$ begins with 1 and the
others begin with 2, or vice versa. There are $3F_{n-1}F_{n-2}^2$ such
triples of the first type. Similarly there are $3F_{n-1}^2F_{n-2}$
of the second type, i.e., one of
$a,b,c$ begins with 2 and the others begin with 1. Hence
begin{eqnarray*} F_n^3 & = & F_{n-1}^3 + F_{n-2}^3
+3(F_{n-1}^2F_{n-2}+F_{n-1}F_{n-2}^2)\ & = &
F_{n-1}^3 + F_{n-2}^3 +3F_{n-1}F_{n-2}(F_{n-1}+F_{n-2})\ & = &
F_{n-1}^3 + F_{n-2}^3 + F_nF_{n-1}F_{n-2}. end{eqnarray*}
answered 11 hours ago
Richard StanleyRichard Stanley
29.1k9115190
29.1k9115190
2
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
3
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
6
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
add a comment |
2
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
3
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
6
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
2
2
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
$begingroup$
With the greatest respect, and mostly out of curiosity, would you really prefer such a bijective proof to the algebra in e.g. Elkies's comment?
$endgroup$
– Lucia
11 hours ago
3
3
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
$begingroup$
If it's simply a matter of proving the identity, then I prefer Elkies. If you want to understand it combinatorially, then a bijective proof is better.
$endgroup$
– Richard Stanley
11 hours ago
6
6
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
$begingroup$
The OP specified a desire for "combinatorial" or "conceptual" explanations. But the distinction between combinatorics and algebra is blurry here. You have to choose one ball from each of three urns, each containing $a$ amaranth and $b$ blue balls. How many choices don't have all three balls the same color? On the one hand, $(a+b)^3-a^3-b^3$. On the other, choose any cyclic permutation of (amaranth, blue, either) to get $3ab(a+b)$.
$endgroup$
– Noam D. Elkies
11 hours ago
add a comment |
$begingroup$
This is just the following identity:
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$ Since $$F_n+(-F_{n-1})+(-F_{n-2})=0,$$ your formula follows.
$endgroup$
7
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
$begingroup$
This is just the following identity:
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$ Since $$F_n+(-F_{n-1})+(-F_{n-2})=0,$$ your formula follows.
$endgroup$
7
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
$begingroup$
This is just the following identity:
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$ Since $$F_n+(-F_{n-1})+(-F_{n-2})=0,$$ your formula follows.
$endgroup$
This is just the following identity:
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$ Since $$F_n+(-F_{n-1})+(-F_{n-2})=0,$$ your formula follows.
answered 15 hours ago
Cherng-tiao PerngCherng-tiao Perng
720147
720147
7
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
$endgroup$
– Noam D. Elkies
15 hours ago
add a comment |
7
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
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– Noam D. Elkies
15 hours ago
7
7
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
$endgroup$
– Noam D. Elkies
15 hours ago
$begingroup$
Simpler yet: since $F_n = F_{n-1} + F_{n+2}$ it's enough to use the two-variable identity $(a+b)^3 - a^3 - b^3 = 3ab(a+b)$ which is a quick consequence of the binomial expansion of $(a+b)^3$.
$endgroup$
– Noam D. Elkies
15 hours ago
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$begingroup$
I get a different left hand side. Rewrite the n+2 term as the sum of n+1 and n terms, and then compute the difference of cubes and divide by three. Algebraically you get the product of the n term and the n+1 term and (the sum of n+1 and n terms). This seems to have more to do with (a+b)^n - a^n - b^n than with Fibonacci. Gerhard "Unsure Of Any Combinatorial Interpretation" Paseman, 2019.03.26.
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– Gerhard Paseman
16 hours ago
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Thanks, edited accordingly.
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– T. Amdeberhan
16 hours ago
5
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$(a+b)^3 - a^3 - b^3 = 3ab(a+b)$. If $a,b$ are consecutive Fibonacci numbers then $a+b$ is the next.
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– Noam D. Elkies
15 hours ago