OFFSET
0,2
COMMENTS
Row sums = A000295: (1, 4, 11, 26, 57, 120, ...).
If we regard the sequence as an infinite square array read by diagonals then it has the formula U(n,k) = (2^n + 2^k)/2 - 1. This appears to coincide with the number of n X k 0..1 arrays colored with only straight tiles, and new values 0..1 introduced in row major order, i.e., no equal adjacent values form a corner. (Fill the array with 0's and 1's. There must never be 3 adjacent identical values making a corner, only same values in a straight line.) Some solutions with n = k = 4 are:
0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1
1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1 0 1 0
1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1
0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1
LINKS
Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
R. H. Hardin, Post to the SeqFan list, Feb 26 2013
FORMULA
T(n,m) = ((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2. - Roger L. Bagula, Oct 16 2008
EXAMPLE
First few rows of the triangle:
1;
2, 2;
4, 3, 4;
8, 5, 5, 8;
16, 9, 7, 9, 16;
32, 17, 11, 11, 17, 32;
64, 33, 19, 15, 19, 33, 64;
128, 65, 35, 23, 23, 35, 65, 128;
...
MATHEMATICA
Table[Table[((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2, {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Oct 16 2008 *)
PROG
(Haskell)
a131816 n k = a131816_tabl !! n !! k
a131816_row n = a131816_tabl !! n
a131816_tabl = map (map (subtract 1)) $
zipWith (zipWith (+)) a130321_tabl a059268_tabl
-- Reinhard Zumkeller, Feb 27 2013
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jul 18 2007
EXTENSIONS
Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar
STATUS
approved