[seqfan] Is this triangle in OEIS?
Richard Guy
rkg at cpsc.ucalgary.ca
Fri Nov 23 23:52:33 CET 2012
Dear all,
A000127 is an old chestnut, as one can see from its
number. But does the triangle associated with it occur
somewhere in OEIS? There's a (quite recent) remark about its being the
sum of the first 5 entries in Pascal's triangle. A usual
derivation of the formula
n choose 4 + n choose 2 + n choose 0
is vie Euler's formula, but we can also get it by a simple
counting of the number of regions by noting that each new
region is formed by a segment of the chord drawn from a new
point to each of the old points in turn.
Start with no points or 1 point and 1 region.
The a second point adds one more region:
1 + 1 = 2
and a third, 2 + 1 + 1 = 4
and a 4th 4 + 1 + 2 + 1 = 8
and a 5th 8 + 1 + 3 + 3 + 1 = 16
and a 6th 16 + 1 + 4 + 5 + 4 + 1 = 31
and a 7th 31 + 1 + 5 + 7 + 7 + 6 + 1 = 57
and an 8th 57 + 1 + 6 + 9 + 10 + 9 + 6 + 1 = 99
and a 9th 99 + 1 + 7 + 11 + 13 + 13 + 11 + 7 + 1 = 163, etc
where the diagonals of the triangle, in either direction,
are APs with common difference 0, 1, 2, 3, ...
Of course, the first 4 lines of the triangle are the same
as in Pascal's triangle, but after that they are in some
sense simpler.
The sums of the rows are the first differences of A000127,
and are the cake numbers, A000125. The second differences
are the central polygonal numbers, A000124, and the third
differences the natural numbers, A000027. R.
More information about the SeqFan
mailing list