Spiral recurrence

Spiral recurrence is a special type of algorithm to make number sequences.

A lot of sequences can be described by a type of recurrence, just like the Fibonacci sequence and the wave sequences described on this site.

Spiral recurrence is a type of recurrence that cannot be described easily with a formule, but it can be described easily by a geometrical interpretation. For that we choose a form of the number spiral to be created - a triangle, a square or a hexagon. The easiest form to type down is the square form.

Square spiral sequence

Below we show the square number spiral, by numbering the positions. Seeing this spiral as a number sequence un, these position numbers give n.

5 4 3

6 1 2 11

7 8 9 10

Of course this spiral can be extended infinitely. We go alternating into four directions, to the right (1), upwards (2), to the left (3) and downward (4). When for instance we are going from position 10 to 11, we go in direction 2. The previous direction is then of course direction 1.

To fill in values, we start with initial value u1 = 1. Each time the next value is found by taking the row of the previous number in the previous direction. By this we get:

4   3   2

9 1 1 94

11 24 28 31

As an example we have made bold the numbers to be added to get the underlined number.

Triangular spiral sequence

The triangular spiral sequence starts as



5 2 101

6 1 1 93

6 >19 22 23 23


The underlined value again is the sum of the bold values.

Hexagonal spiral sequence

For the hexagonal spiral we make use of a dummy center point. Below the first part is again given.


5 3

8 * 2 110

10 1 1 98

23 33 42

The best collection of integer sequences is Neil Sloane's On-Line Encyclopedia of Integer Sequences.

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