A168314 Eigensequence of triangle A168313.

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

Author

Gary W. Adamson, Nov 22 2009

Keywords

• eigen
• nonn

Comments

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

Crossrefs

Cf. A168313, A168315.

Programs

Python

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

Formula

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.

Extensions

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