Template:Sequence of the Day for August 7

Intended for: August 7, 2011

Timetable

• First draft entered by Alonso del Arte on May 7, 2011 based on a write-up by Charles R. Greathouse. ✓
• Second draft to be entered by May 27, 2011 by which time A109794 will hopefully have been edited with a proof or disproof of its periodicity.
• Draft reviewed by Daniel Forgues on August 6, 2011 ✓, August 6, 2018 ✓
• Draft to be approved by July 7, 2011

Yesterday's SOTD * Tomorrow's SOTD

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A109794:

a (2 n) =

A001906

(n + 1)

,

a (2 n + 1) =

A002878

(n), n   ≥   0

. Equivalently,

a (2 n) =

[even index] bisection A000045

(2 n + 2)

of Fibonacci numbers interleaved with

a (2 n + 1) =

[odd index] bisection A000204

(2 n + 1)

of Lucas numbers.

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

This sequence, on the face of it, is just another linear recurrence relation (of order 4), e.g.

a (0) = 1, a (1) = 1, a (2) = 3, a (3) = 4;

a (n) = 3 a (n − 2) − a (n − 4), n ≥ 4.

It appears (proof?) that A109794

(n) =

A189761

(n + 2), n   ≥   4,

where A189761 gives numbers

n

for which the set of residues

{Fibonacci (k) mod n, k = 0, 1, 2, ...}

is minimal, i.e.

n > m ⟹

A066853

(n) >

A066853

(m)

. It is also conjectured that the members of the sequence, for

n   ≥   4

, are just those numbers for which the Pisano period is minimal: that is, conjecturally, A001175

(n) >

A001175

(m)

for all

n > m

iff

m

is in this sequence.