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