A085348 Ratio-determined insertion sequence I(0.264) (see the link below).

1, 4, 19, 72, 341, 1292, 6119, 23184, 109801, 416020, 1970299, 7465176, 35355581, 133957148, 634430159, 2403763488, 11384387281, 43133785636, 204284540899, 774004377960, 3665737348901, 13888945017644, 65778987739319

Offset: 0

Author

John W. Layman, Jun 24 2003

Keywords

• nonn

Comments

This is one of the "twin" ratio-determined insertion sequences (RDIS) that are "children" in the next generation below the "parent" sequences I(0.25024) (A004253) and I(0.26816) (A001353) in the recurrence tree of RDIS sequences. The RDIS twin of this sequence is A085349. See the link for an explanation of RDIS twin. See A082630 or A082981 for other recent examples of RDIS sequences.

Assuming that a(n) = 18a(n-2) - a(n-4) is true: For n >= 2, a(n) = (t(i+2n+2) - t(i))/(t(i+n+2) + t(i+n)*(-1)^(n-1)), where (t) is any recurrence of the form (4,1) without regard to initial values. With an additional initional 0 is this sequence the union of A060645 for even n and A049629 for odd n. - Klaus Purath, Sep 22 2024

Links

• John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types, June 2003 [Broken link]
• John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types, June 2003 [local copy, corrected]
• John W. Layman, Sequences Generated by Age-Determined Insertion Trees, Jan 2006
• John W. Layman, Sequences Generated by Age-Determined Insertion Trees, Jan 2006 [Local copy]

Crossrefs

Cf. A001353, A004253, A082630, A082981, A085349.

Formula

It appears that a(n)=18a(n-2)-a(n-4).

Apparently a(n)a(n+3) = -4 + a(n+1)a(n+2). - Ralf Stephan, May 29 2004

From Klaus Purath, Sep 22 2024: (Start)

Assuming that a(n) = 18a(n-2) - a(n-4) is true:

a(2n) = 5a(2n-1) - a(2n-2), n >= 1.

a(2n+1) = 4a(2n) - a(2n-1), n >= 1. (End)