A168313 -id:A168313 - 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.

A168314 Eigensequence of triangle A168313.

Original entry on oeis.org

1, 1, 3, 5, 13, 29, 71, 165, 401, 957, 2315, 5561, 13437, 32377, 78191, 188617, 455425, 1099137, 2653699, 6405733, 15465165, 37334149, 90133463, 217596445, 525326353, 1268238029, 3061802411, 7391815977, 17845434365, 43082619953, 104010674271, 251103812113

Offset: 1

Gary W. Adamson, Nov 22 2009

Conjectured convergent of a(n)/a(n-1) = (1+sqrt(2)). E.g.: a(19)/a(18) = 2653699/1099137 = 2.4143478...

Cf. A168313, A168315.

a = [1]
for n in range(30):
    a.append(2*sum(a[n//2:-1]) + a[-1])
print(a) # Andrey Zabolotskiy, Aug 28 2024

Equals the eigensequence of triangle A168313 = lim_{n->oo} M^n where M = A168313 shifted down one row and inserting a "1" at top.

Equals the left column vector as a sequence.

Terms a(20) and beyond from Andrey Zabolotskiy, Aug 28 2024

cp's OEIS Frontend

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

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