

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;...).


6



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
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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)/21. This appears to coincide with the number of nXk 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 are:
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, A000295.
Sequence in context: A157927 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



