Question

$gcd(f_k, f_{k+3})$, where $f_k$ is the k'th Fibonacci number

score 133 · Answer 1 · 2016-12-12 15:36:41

133

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$gcd(f_k, f_{k+3})$, where $f_k$ is the k'th Fibonacci number

Published on 12 Dec 2016 - 15:36

score 152 · Answer 2 · 2026-03-25 23:37:24

152

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Another simple rule satisfied by the Fibonacci $n$-step constants?

Published on 25 Mar 2026 - 23:37

score 355 · Answer 3 · 2026-03-25 22:05:02

355

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The famous root of $x(1 - x^{12})^2(1 - x^{24})^2 = (1 - x^6)^7(1 - x^8)^4$

Published on 25 Mar 2026 - 22:05

score 928 · Answer 4 · 2016-12-16 02:24:05

928

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Prove Fibonacci Identity using generating functions

Published on 16 Dec 2016 - 2:24

score 195 · Answer 5 · 2016-12-19 07:37:56

195

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Show that ${(F_n^2+F_{n+1}^2+F_{n+2}^2)^2\over F_{n}^4+F_{n+1}^4+F_{n+2}^4}=2$

Published on 19 Dec 2016 - 7:37

score 117 · Answer 6 · 2016-12-20 13:18:29

117

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Help to prove $F_n^5+F_{n+1}^5=F_{n+2}[(F_nF_{n+1}+F_{n-1}^2)^2+F_{n-1}^2F_nF_{n+1}]$

Published on 20 Dec 2016 - 13:18

score 233 · Answer 7 · 2016-12-21 18:54:04

233

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Proof that $\sum\limits_{k=1}^{n}(n+1-k)^2F_k^2=F_{n+1}(F_n+F_{n+2})-n-1$

Published on 21 Dec 2016 - 18:54

score 1k · Answer 8 · 2016-12-22 05:53:41

1k

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Fibonacci in Arithmetic Progression

Published on 22 Dec 2016 - 5:53

score 139 · Answer 9 · 2016-12-23 07:44:32

139

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Show that $\arctan\left({1\over A}\right)=\arctan\left({1\over B}\right)+\arctan\left({1\over C}\right)$

Published on 23 Dec 2016 - 7:44

score 154 · Answer 10 · 2016-12-23 10:31:07

154

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Show that $S_L= L_n^2L_{n+1} -L_1L_2L_3-L_2L_3L_4-\cdots L_{n-2}L_{n-1}L_n$, $n\ge3$

Published on 23 Dec 2016 - 10:31

score 186 · Answer 11 · 2016-12-24 07:38:22

186

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How can we show that $2^3+1^3+3^3+4^3+\cdots+L_n^3={L_nL_{n+1}^2+5(-1)^n[L_{n-1}+2(-1)^n]+9\over 2}$

Published on 24 Dec 2016 - 7:38

score 185 · Answer 12 · 2026-03-25 22:06:09

185

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Prove that:$(\phi-1)^1(2-\phi)^{1\over4}(2\phi-3)^{1\over9}(5-3\phi)^{1\over16}\cdots(F_{n+1}-\phi F_n)^{1\over n^2}=\phi^{-H_n}$

Published on 25 Mar 2026 - 22:06

score 89 · Answer 13 · 2016-12-27 15:05:34

89

Views

Is there a closed form or series representation for this linear recurrence?

Published on 27 Dec 2016 - 15:05

score 930 · Answer 14 · 2017-01-02 15:39:14

930

Views

Why there is a geometrical boundary condition on the rotated phi/fibonacci spiral?

Published on 02 Jan 2017 - 15:39

score 867 · Answer 15 · 2017-01-03 23:53:26

867

Views

Fibonacci Sequence explained to a noob

Published on 03 Jan 2017 - 23:53

List Question

$gcd(f_k, f_{k+3})$, where $f_k$ is the k'th Fibonacci number

Another simple rule satisfied by the Fibonacci $n$-step constants?

The famous root of $x(1 - x^{12})^2(1 - x^{24})^2 = (1 - x^6)^7(1 - x^8)^4$

Prove Fibonacci Identity using generating functions

Show that ${(F_n^2+F_{n+1}^2+F_{n+2}^2)^2\over F_{n}^4+F_{n+1}^4+F_{n+2}^4}=2$

Help to prove $F_n^5+F_{n+1}^5=F_{n+2}[(F_nF_{n+1}+F_{n-1}^2)^2+F_{n-1}^2F_nF_{n+1}]$

Proof that $\sum\limits_{k=1}^{n}(n+1-k)^2F_k^2=F_{n+1}(F_n+F_{n+2})-n-1$

Fibonacci in Arithmetic Progression

Show that $\arctan\left({1\over A}\right)=\arctan\left({1\over B}\right)+\arctan\left({1\over C}\right)$

Show that $S_L= L_n^2L_{n+1} -L_1L_2L_3-L_2L_3L_4-\cdots L_{n-2}L_{n-1}L_n$, $n\ge3$

How can we show that $2^3+1^3+3^3+4^3+\cdots+L_n^3={L_nL_{n+1}^2+5(-1)^n[L_{n-1}+2(-1)^n]+9\over 2}$

Prove that:$(\phi-1)^1(2-\phi)^{1\over4}(2\phi-3)^{1\over9}(5-3\phi)^{1\over16}\cdots(F_{n+1}-\phi F_n)^{1\over n^2}=\phi^{-H_n}$

Is there a closed form or series representation for this linear recurrence?

Why there is a geometrical boundary condition on the rotated phi/fibonacci spiral?

Fibonacci Sequence explained to a noob

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