Template:Sequence of the Day for July 7

Intended for: July 7, 2012

Timetable

• First draft entered by Alonso del Arte on March 12, 2011 based on an almost verbatim copy of a write-up by Daniel Forgues from November 12, 2010.
• First draft rejected because it is already being used for Sequence of the Day for July 2.
• New first draft entered by Alonso del Arte on April 30, 2011 mostly just copying comments made by John Blythe Dobson. ✓
• Draft reviewed by Daniel Forgues on May 5, 2011 ✓
• Draft approved by Alonso del Arte on July 6, 2011 ✓

Yesterday's SOTD * Tomorrow's SOTD

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A000032: Lucas numbers (beginning with 2)*:

L (n) = L (n  −  1) + L (n  −  2)

with

L (0) = 2, L (1) = 1

.

{ 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... }

John Blythe Dobson connected some dots regarding Lucas and Fibonacci numbers:

The comments submitted by Miklos Kristof from Mar 19, 2007 for the Fibonacci numbers (A000045) contain four important identities which have close analogues in the Lucas numbers (A000204): For

a   ≥   b

and odd

b

,

L (a + b) + L (a  −  b) = 5 F (a)  F (b)

. For

a   ≥   b

and even

b

,

L (a + b) + L (a  −  b) = L (a)  L (b)

. For

a   ≥   b

and odd

b

,

L (a + b)  −  L (a  −  b) = L (a)  L (b)

. For

a   ≥   b

and even

b

,

L (a + b)  −  L (a  −  b) = 5 F (a)  F (b)

.

_______________

* A000204 Lucas numbers (beginning with 1):

L (n) = L (n  −  1) + L (n  −  2)

with

L (1) = 1, L (2) = 3

.