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

0,2

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

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

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

Cf. A130321, A059268, A000012.

Row sums give A000295(n+2).

Sequence in context: A374396 A227256 A328774 * A223541 A128181 A125185

Adjacent sequences: A131813 A131814 A131815 * A131817 A131818 A131819

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