A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

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, 256, 129, 67, 39, 31, 39, 67, 129, 256, 512, 257, 131, 71, 47, 47, 71, 131, 257, 512, 1024, 513, 259, 135, 79, 63, 79, 135, 259, 513, 1024

Offset: 0

Author

Gary W. Adamson, Jul 18 2007

Keywords

• nonn
• tabl

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

(Observation from R. H. Hardin, cf. link.) - M. F. Hasler and N. J. A. Sloane, Feb 26 2013

Examples

			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;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• R. H. Hardin, Post to the SeqFan list, Feb 26 2013

Crossrefs

Cf. A130321, A059268, A000012.

Row sums give A000295(n+2).

Programs

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

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 *)

Formula

T(n,m) = ((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2. - Roger L. Bagula, Oct 16 2008

Extensions

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar