A168314 -id:A168314 - cp's OEIS frontend

A168315 Triangle read by rows, A168313 * the diagonalized variant of its eigensequence, A168314.

1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 0, 0, 2, 6, 5, 0, 0, 0, 6, 10, 13, 0, 0, 0, 6, 10, 26, 29, 0, 0, 0, 0, 10, 26, 58, 71, 0, 0, 0, 0, 10, 26, 58, 142, 165, 0, 0, 0, 0, 0, 26, 58, 142, 330, 401, 0, 0, 0, 0, 0, 26, 58, 142, 330, 802, 957

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401, 957,...).
Rightmost column = A168314 prefaced with a 1.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
0, 1;
0, 2, 1;
0, 0, 2, 3;
0, 0, 2, 6, 5;
0, 0, 0, 6, 10, 13;
0, 0, 0, 6, 10, 26, 29;
0, 0, 0, 6, 10, 26, 58, 71;
0, 0, 0, 0, 10, 26, 58, 142, 165;
0, 0, 0, 0, 0, 26, 58, 142, 330, 401;
0, 0, 0, 0, 0, 26, 58, 142, 330, 802, 957;
0, 0, 0, 0, 0, 0, 58, 142, 330, 802, 1914, 2315;
0, 0, 0, 0, 0, 0, 58, 142, 330, 802, 1914, 4630, 5561;
...

Cf. A168313, A168314

Let M = triangle A168313 and Q = in an infinite lower triangular matrix with
A168314 prefaced with a 1 as the rightmost diagonal with the rest of terms 0's.
Triangle A168315 = M*Q.

A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = odd integers repeated: (1, 1, 3, 3, 5, 5,...).

Eigensequence of the triangle = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401,...).

			First few rows of the triangle =
1;
0, 1;
0, 2, 1;
0, 0, 2, 1;
0, 0, 2, 2, 1;
0, 0, 0, 2, 2, 1;
0, 0, 0, 2, 2, 2, 1;
0, 0, 0, 0, 2, 2, 2, 1;
0, 0, 0, 0, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
...

Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A101688, A168314, A168315

rows = 11;
A = Array[Which[#1 == 1, 1, #1 <= #2, 2, True, 0]&, {rows, rows}];
Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, Aug 08 2018 *)

Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
From Boris Putievskiy, Jan 09 2013: (Start)
a(n) = 2*A101688(n)-A023531(n).
a(n) = 2*floor((2*A002260(n)+1)/(A003056(n)+3))*A002260(n)-A023531(n).
a(n) = 2*floor((2*n-t*(t+1)+1)/(t+3))*(n-t*(t+1)/2) - floor((sqrt(8*n+1)-1)/2) + t, where t = floor((-1+sqrt(8*n-7))/2). (End)

cp's OEIS Frontend

A168315 Triangle read by rows, A168313 * the diagonalized variant of its eigensequence, A168314.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula