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Template:Sequence of the Day for November 7

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Intended for: November 7, 2012

Timetable

  • First draft entered by Mitch Harris on July 29, 2011
  • Draft reviewed by Alonso del Arte on April 4, 2012
  • Draft approved by Daniel Forgues on November 7, 2012
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



A002620: Quarter-squares:
n
 2
n
 2
, n   ≥   0
. Equivalently,
n 2
4
, n   ≥   0
.
{ 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100... }

This sequence is fairly simple, just the interleaving of two already simple sequences, namely the squares (A000290) and the pronic numbers (A002378), i.e. the [even index/odd index] bisections of this sequence are

     
a (2 n)  =  n 2, n ≥ 0,
a (2 n + 1)  =  n  (n + 1)  =  2  tn , n ≥ 0,

where
tn
is the
n
th triangular number.

The recurrences for the [even index/odd index] bisections are

A000290 (0) = 0; A000290 (n) = A000290 (n  −  1) + 2 n  −  1, n   ≥   1, and
A002378 (0) = 0; A002378 (n) = A002378 (n  −  1) + 2 n, n   ≥   1,

which leads to the simple recurrence

A002620 (0) = 0; A002620 (n) = A002620 (n  −  1) +
n
 2
, n   ≥   1,
where
n
 2
is given by A004526.
Suppose you try to fill an
n  ×  n
discrete grid, node by node, keeping what is covered as compact and simple as possible. If you start with a square covering
n  ×  n
nodes, you’ll add one column (of
n
nodes) covering
n  ×  (n + 1)
nodes, then add another row (of
n + 1
nodes), to now cover
(n + 1)  ×  (n + 1)
nodes. Covering nodes in this ‘herring-bone’ pattern generates the quarter squares.