OFFSET
1,2
COMMENTS
From Timothy L. Tiffin, Aug 15 2021: (Start)
Each a(n) also gives the number of 2-combinations from the set {1,2,3,...,n+1} that agree when written in (a) lexicographic order and (b) colexicographic order. For example, the 15 2-combinations from the set {1,2,3,4,5,6} using each order (with ** indicating agreement) would be:
(a) 12 13 14 15 16 23 24 25 26 34 35 36 45 46 56
(b) 12 13 23 14 24 34 15 25 35 45 16 26 36 46 56
** ** ** ** **
So, a(5) = 5. A way to visualize this correspondence is to take a 5 X 5 matrix with columns labeled 1, 2, 3, 4, 5 and rows labeled 2, 3, 4, 5, 6 and construct a left-justified lower-triangular array of 2-combinations as seen in the first diagram below:
1 2 3 4 5 1 2 3 4 5
--------------- -----------
2| 12 2| x
3| 13 23 3| x o
4| 14 24 34 4| o o o
5| 15 25 35 45 5| o x o o
6| 16 26 36 46 56 6| o o o x x
Now, traversing through this triangular array (a) downwards or (b) across will respectively generate the lexicographic ordering or the colexicographic ordering seen above. In the second diagram above, "x" indicates where the 2-combinations agree and "o" indicates where they don't. (End)
FORMULA
For n>1, a(n) = 2 + d(2n^2 - 6n + 5), where d(k) is number of divisors of k. - Dean Hickerson
EXAMPLE
For n=1, (a) = 1, (b) = 1, so a(1)=1.
For n=3, (a) = 1; 2 4; 3 5 6, (b) = 1; 2 3; 4 5 6, so a(3)=4.
MATHEMATICA
Join[{1}, Table[2+DivisorSigma[0, 2n^2-6n+5], {n, 2, 130}]] (* Harvey P. Dale, Jan 12 2022 *)
PROG
(PARI) a(n) = if (n==1, 1, 2 + numdiv(2*n^2 - 6*n + 5)) \\ Michel Marcus, Jun 15 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
James Propp, Dec 11 1999
EXTENSIONS
More terms from David Radcliffe
"left-justified lower" added to the Name by Timothy L. Tiffin, Aug 15 2021
STATUS
approved