Sequence Numbers: Combinatorial Functions
By David Brooks
C (n,2): The number of ways to choose two
items from a group of n items. (This
sequence is also called the Triangular numbers, and the third diagonal in
Pascal’s Triangle. At the end of this
post I will show how this sequence is related the 3, 3, 1 Tribonacci
sequence.)
The Sequence Number is: 999,997,000,002,999,999
1/999997000002999999 =
0.
000000 000000 000001 000003 000006 000010 000015 000021 000028 000036 000045 000055 000066 000078 …
Terms are written in six digit strings.
Compare with OEIS sequence A000217.

C (n, 3): The number of ways to choose
three items from a group of n items.
(This sequence is also called the Tetrahedral numbers and the fourth
diagonal in Pascal’s Triangle.)
The Sequence Number is: 999,996,000,005,999,996,000,
001
1/999996000005999996000
001 =
0.
000000 000000 000000 000001 000004 000010 000020 000035 000056 000084 000120 000165 000220 000286 …
Terms are written in six digit strings.
Compare with OEIS sequence A000292.

C (n, 4): The number of ways to choose
four items from a group of n items.
The Sequence Number is: 999,995,000,009,999,990,000,
004,999,999
1/9999950000099999900000
04999999 =
0.
000000 000000 000000 000000 000001 000005 000015 000035 000070 000126 000210 000330 000495 000715 001001 001365 001820 …
Terms are written in six digit strings.
These terms are also the numbers found in
the fifth diagonal of Pascal’s Triangle.
Compare with OEIS sequence A000332.

C (n, 5): The number of ways to choose
five items from a group of n items.
The Sequence Number is: 999,994,000,014,999,980,000,
014,999,994,000,001
1/9999940000149999800000
14999994000001 =
0.
000000 000000 000000 000000 000000 000001 000006 000021 000056 000126 000252 000462 000792 001287 002002 003003 004368 …
Terms are written in six digit strings.
These terms are also the numbers found in
the sixth diagonal of Pascal’s Triangle.
Compare with OEIS sequence A000389.

C (n, 6): The number of ways to choose
six items from a group of n items.
The Sequence Number is: 999993000020999965000034
999979000006999999
1/9999930000209999650000
34999979000006999999 =
0.
000000 000000 000000 000000 000000 000000 000001 000007 000028 000084 000210 000462 000924 001716
Terms are written in six digit strings.
These terms are also the numbers found in
the seventh diagonal of Pascal’s Triangle.
Compare with OEIS sequence A000579.

C (n, 7):
The number of ways to choose seven items from a group of n items.
The Sequence Number is:
9999920000279999440000699 99944000027999992000001
1/99999200002799994400006
9999944000027999992000001=
0.
000000 000000 000000 000000 000000 000000 000000 000001 000008 000036 000120 000330 000792 001716 …
Terms are written in six digit strings.
These terms are also the numbers found in
the eighth diagonal of Pascal’s Triangle.
Compare with OEIS sequence A000580.

C (n, 8): The number of ways to choose
eight items from a group of n items.
The Sequence number is:
999991000035999916000125 999874000083999964000008 999999
1/99999100003599991600012
599987400008399996400000 8999999 =
0,
000000 000000 000000 000000 000000 000000 000000 000000 000001 000009 000045 000165 000495 001287 003003 006435 012870 024310 043758 075582 …
Terms are written in six digit strings.
These terms are also the numbers found in
the ninth diagonal of Pascal’s Triangle.
Compare with OEIS sequence A000581.

C (n, 9): The number of ways to choose
nine items from a group of n items.
The Sequence Number is:

If you recall, 999,997,000,002,999,999
was the Sequence Number we used to produce the terms of C (n,2). But this sequence number looks like a
Sequence number that we would use to produce a tribonacci like sequence.
If we interpret 999,997,000,002,999,999
as a Sequence Number for a Tribonacci like sequence it translates into a
sequence defined as: a(0) = a(1) = 0, a(2) = 1, and when n>1 then a(n) = 3*a(n1)
– 3*a(n2) + a(n3). This would be
called the 3, 3, 1 Tribonacci sequence.
I set up a spreadsheet to calculate the
terms of this sequence, and printed the results below. I calculated the terms using the recursive
formula at the end of the paragraph above.
You can compare the results with the results from the Sequence Number
for C (n, 3).
So C (n, 2), Triangular Numbers, the
third diagonal of Pascal’s Triangle, and the 3, 3, 1 Tribonacci sequence are
mathematically related.
I think it is amazing how many branches
of mathematics have a relationship to Pascal’s Triangle. I guess I need to do a series of posts in
the near future about Pascal’s Triangle.

David
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