Question

Proof of $f(x) = \sum_{k=0}^\infty a_kx^k$ with $a_0 := a_1 := 1$ and $a_{n+2} := a_n+a_{n+1}$

score 79 · Answer 1 · 2019-12-04 20:21:19

79

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Proof of $f(x) = \sum_{k=0}^\infty a_kx^k$ with $a_0 := a_1 := 1$ and $a_{n+2} := a_n+a_{n+1}$

Published on 04 Dec 2019 - 20:21

score 176 · Answer 2 · 2019-12-07 12:02:19

176

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proof : even nth Fibonacci number using Mathematical Induction

Published on 07 Dec 2019 - 12:02

score 39 · Answer 3 · 2019-12-10 10:41:02

39

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Why is the fibonacci sequence generated in this "game"?

Published on 10 Dec 2019 - 10:41

score 172 · Answer 4 · 2026-03-27 11:49:00

172

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Fibonacci sum for $\pi$: $\sum_{n=1}^\infty\frac{F_{2n}}{n^2\binom{2n}{n}}=\frac{4\pi^2}{25\sqrt5}$

Published on 27 Mar 2026 - 11:49

score 1.2k · Answer 5 · 2019-12-12 03:26:24

1.2k

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Staircase recursion problem using recurrence relations.

Published on 12 Dec 2019 - 3:26

score 72 · Answer 6 · 2019-12-14 19:18:28

72

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the sum of $f_0+f_2+f_4+f_6+\cdots+f_{2n}$

Published on 14 Dec 2019 - 19:18

score 117 · Answer 7 · 2019-12-18 15:08:17

117

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Fibonacci terms coincide with polynomial

Published on 18 Dec 2019 - 15:08

score 67 · Answer 8 · 2019-12-19 09:26:26

67

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Every natural number can be represented as a sum of Fibonacci numbers.

Published on 19 Dec 2019 - 9:26

score 99 · Answer 9 · 2019-12-26 02:27:47

Prove that for all natural $n$, $F_{2n - 1}^2 \equiv -1 \pmod {F_{2n + 1}}$ and $F_{2n + 1}^2 \equiv -1 \pmod {F_{2n - 1}}$.

score 44 · Answer 10 · 2019-12-29 15:22:54

44

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recursive relation for generating sequence whose ratio converges to phi+k, where phi is golden ratio

Published on 29 Dec 2019 - 15:22

score 123 · Answer 11 · 2019-12-30 06:02:23

Given $F_m$ be the $m^\text{th}$ number in the Fibonacci sequence. Prove that for all natural $n$, $|F_n^2 + F_nF_{n + 1} - F_{n + 1}^2| = 1$.

score 70 · Answer 12 · 2019-12-30 15:47:53

70

Views

Calculate $\sum_{i=2}^{∞} i\,\frac{\operatorname{fib}(i-1)}{2^i}$

Published on 30 Dec 2019 - 15:47

score 102 · Answer 13 · 2020-01-02 06:40:40

102

Views

A new pattern of numbers that repeats itself which is related to Fibonacci sequence . Is it the shortest one?

Published on 02 Jan 2020 - 6:40

score 108 · Answer 14 · 2020-01-03 16:41:39

108

Views

Fibonacci sequence but switching between adding and subtracting

Published on 03 Jan 2020 - 16:41

score 209 · Answer 15 · 2020-01-05 21:16:47

209

Views

Is this sequence based on the Fibonacci numbers a prime generator?

Published on 05 Jan 2020 - 21:16

List Question

Proof of $f(x) = \sum_{k=0}^\infty a_kx^k$ with $a_0 := a_1 := 1$ and $a_{n+2} := a_n+a_{n+1}$

proof : even nth Fibonacci number using Mathematical Induction

Why is the fibonacci sequence generated in this "game"?

Fibonacci sum for $\pi$: $\sum_{n=1}^\infty\frac{F_{2n}}{n^2\binom{2n}{n}}=\frac{4\pi^2}{25\sqrt5}$

Staircase recursion problem using recurrence relations.

the sum of $f_0+f_2+f_4+f_6+\cdots+f_{2n}$

Fibonacci terms coincide with polynomial

Every natural number can be represented as a sum of Fibonacci numbers.

Prove that for all natural $n$, $F_{2n - 1}^2 \equiv -1 \pmod {F_{2n + 1}}$ and $F_{2n + 1}^2 \equiv -1 \pmod {F_{2n - 1}}$.

recursive relation for generating sequence whose ratio converges to phi+k, where phi is golden ratio

Given $F_m$ be the $m^\text{th}$ number in the Fibonacci sequence. Prove that for all natural $n$, $|F_n^2 + F_nF_{n + 1} - F_{n + 1}^2| = 1$.

Calculate $\sum_{i=2}^{∞} i\,\frac{\operatorname{fib}(i-1)}{2^i}$

A new pattern of numbers that repeats itself which is related to Fibonacci sequence . Is it the shortest one?

Fibonacci sequence but switching between adding and subtracting

Is this sequence based on the Fibonacci numbers a prime generator?

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