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# Integer Sequence K-12 (Banff, 2015)

Report on Integer Sequences K-12 Conference

(held at the Banff International Research Station Feb. 27th – March 1st, 2015)

Web site for the conference:

http://www.birs.ca/events/2015/2-day-workshops/15w2178/

Videos for the conference:

http://www.birs.ca/events/2015/2-day-workshops/15w2178/videos/

## Summary from Gordon Hamilton, March 21 2015

The following sequences were selected by the mathematicians and educators who attended the Integer Sequences K-12 conference (held at the Banff International Research Station Feb. 27th – March 1st, 2015). The sequences still require to be worked up so that they are useful for teachers, so consider this a sneak-peek. A more thorough peek is to be had in the first 14 pages of the "Integer Sequences Slide Show" report which is stored on Google Docs. If you would like access to this report, please contact Gordon Hamilton (gord at mathpickle dot com) or N. J. A. Sloane (njasloane at gmail dot com). Also, if you want to be part of the process to perfect and communicate these integer sequences, send email saying that you're interested.

## Kindergarten

A034326 Hours struck by a clock.

 { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... }

## Primary education

A030227 Number of n-celled polyominoes with bilateral symmetry.

 { 1, 1, 2, 3, 6, 10, 20, 34, 70, 121, 250, 441, 912, 1630, 3375, ... }

A243205 The Nasty Mr. Sneeze - Consider the n X n Go board as a graph; remove i nodes and let j be the number of nodes in the largest connected subgraph remaining; then a(n) = minimum (i + j).

 { 1, 3, 5, 9, 12, 16, 20, 25, 29, 36?, 41?, 47?, ... }

A254873 Recamán [division, -, +, *] - Starting at the seed number (14) the sequence continues by dividing, subtracting, adding or multiplying by the step number (2). Division gets precedence over subtraction which gets precedence over addition which gets precedence over multiplication. The new number must be a positive integer and not previously listed. The sequence terminates if this is impossible.

 { 14, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 24, 22, 11, 9, 18, 16, 32, 30, 15, 13, 26, 28, 56, 54, 27, 25, 23, 21, 19, 17, 34, 36, 38, 40, 20 }
The sequence terminated!

A253472 Numbers n such that 1, 2, ..., 2n can be partitioned into n pairs, where each pair adds up to a perfect square.

 { 4, 7, 8, 9, 12, 13, 14, 15, 16, ... }

<this sequence beats both A071983 and A252897 a(n) = number of ways to pair the integers 1 to 2n so that the sum of each pair is a square. The latter is a more interesting sequence, but most grade 4s will appreciate the much less scary looking sequence above - Thank you Daniel Finkel for this final observation.>

A256174 Boomerang Fractions - Starting with 1, on the first step add 1/n, and on subsequent steps either add 1/n or take the reciprocal. a(n) = number of steps required to return to 1. (The sequence starts with a(2).)

 { 4, 9, 7, 20, 6, 33, 13, 23, 16, 62?, 8, 75?, 18, 17, 25, ... }

A125508 Integral Fission - A prime factorization tree in which every pair of children is chosen so they are as equal as possible and the largest child goes on the right. a(n) are the lowest numbers for which a new tree shape is encountered.

 { 2, 4, 8, 16, 20, 32, 40, 64, 72, 88, 128, 160, 176, 200, 220, 256, 272, 288, 320, 336, 360, 400, 420, 460, 480, 512, 540, 544, 640, 704, 864, 880, 920, ... }

## Secondary education

A039834 Fibonacci numbers (A000045) extended to negative indices. F(0) = 0 in the sequence below.

 { 1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, ... }

Similar to A226595 Lengths of maximal non touching increasing paths in n X n grids.

 { 0, 2, 4, 7, 9, 12, 15, 17, 20, ... }

... but actually "Lengths of maximal non touching increasing paths in n X n grids starting at the upper left and ending at the lower right."

 { 0, 2, 4, 6, 9, 12?, 15?, 17?, 20, ... }

A069283 The number of ways that n can be written as the sum of at least two consecutive positive integers.

 { 0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 0, 3, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 1, 5, ... }

A225745 Smallest k such that n numbers can be picked in {1,...,k} with no four in arithmetic progression.

 { 1, 2, 3, 5, 6, 8, 9, 10, 13, 15, 17, 19, 21, 23, 25, 27, 28, 30, 33, 34, 37, 40, ... }

A000108 Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

 { 1, 1, 2, 5, 14, 42, 132, 429, 1430, ... }

A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

 { 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... }

## General information

Think of them as classroom ambassadors for The On-Line Encyclopedia of Integer Sequences ;-)

## Reports from the Conference

1. Gordon Hamilton et al., Integer Sequences K-12 (Banff 2015) (32 pages)
2. Gordon Hamilton et al., Additional Notes on Sequences Considered at Banff Conference (54 pages)
3. Henri Piciotto's 10-page report. These are classroom-tested activities related to sequences proposed for inclusion on the above list. Some were adopted, some not. Here is a video of Henri summarizing those activities at the Banff meeting.