GCD of all sums of n consecutive Fibonacci numbers

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A210209 GCD of all sums of 

n

consecutive Fibonacci numbers, 

n   ≥   0

.

 

{0, 1, 1, 2, 1, 1, 4, 1, 3, 2, 11, 1, 8, 1, 29, 2, 21, 1, 76, 1, 55, 2, 199, 1, 144, 1, 521, 2, 377, 1, 1364, 1, 987, 2, 3571, 1, 2584, 1, 9349, 2, 6765, 1, 24476, 1, 17711, 2, ...}

Note that the sum of zero numbers gives the empty sum, defined as the additive identity, i.e. 0. Also, since 0 is divisible by any nonzero integer, it has no greatest divisor: this fact is represented by assigning 0 to the GCD of 0, since 0 / 0 is undefined.

GCD of all sums of odd n consecutive Fibonacci numbers

A131534 GCD of all sums of 

2n  +  1

consecutive Fibonacci numbers, 

n   ≥   0

.

 

{1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...}

Thus 

a (n) = 1  +  [3 ∣ (2n  +  1)]

, 

n   ≥   0

, where 

[·]

is Iverson bracket notation.

GCD of all sums of even n consecutive Fibonacci numbers

A005013 GCD of all sums of 

2n

consecutive Fibonacci numbers, 

n   ≥   0

.

 

{0, 1, 1, 4, 3, 11, 8, 29, 21, 76, 55, 199, 144, 521, 377, 1364, 987, 3571, 2584, 9349, 6765, 24476, 17711, 64079, 46368, 167761, 121393, 439204, 317811, 1149851, 832040, 3010349, 2178309, ...}

For even 

n

, 

a (n) = Fn

; for odd 

n

, 

a (n) = Ln

.

GCD of all sums of n consecutive Lucas numbers

A229339 GCD of all sums of 

n

consecutive Lucas numbers, 

n   ≥   1

.

 

{1, 1, 2, 5, 1, 4, 1, 15, 2, 11, 1, 40, 1, 29, 2, 105, 1, 76, 1, 275, 2, 199, 1, 720, 1, 521, 2, 1885, 1, 1364, 1, 4935, 2, 3571, 1, 12920, 1, 9349, 2, 33825, 1, 24476, 1, 88555, 2, 64079, 1, 231840, 1, ...}

GCD of all sums of odd n consecutive Lucas numbers

A131534 GCD of all sums of 

2n  +  1

consecutive Lucas numbers, 

n   ≥   0

.

 

{1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...}

Thus 

a (n) = 1  +  [3 ∣ (2n  +  1)]

, 

n   ≥   0

, where 

[·]

is Iverson bracket notation.

GCD of all sums of even n consecutive Lucas numbers

A203976 GCD of all sums of 

2n

consecutive Lucas numbers, 

n   ≥   0

.

 

{0, 1, 5, 4, 15, 11, 40, 29, 105, 76, 275, 199, 720, 521, 1885, 1364, 4935, 3571, 12920, 9349, 33825, 24476, 88555, 64079, 231840, 167761, 606965, 439204, 1589055, 1149851, 4160200, ...}

For even 

n

, 

a (n) = A201157(1  +  n / 2)

; for odd 

n

, 

a (n) = Ln

.