A085260 Ratio-determined insertion sequence I(0.0833344) (see the link below).

1, 12, 155, 2003, 25884, 334489, 4322473, 55857660, 721827107, 9327894731, 120540804396, 1557702562417, 20129592507025, 260127000028908, 3361521407868779, 43439651302265219, 561353945521579068

Offset

1,2

Comments

This sequence is the ratio-determined insertion sequence (RDIS) "twin" to A078362 (see the link for an explanation of "twin"). See A082630 or A082981 for recent examples of RDIS sequences.

a(n) = L(n,13), where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005

For n >= 2, a(n) equals the permanent of the (2n-2) X (2n-2) tridiagonal matrix with sqrt(11)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

Seems to be positive values of x (or y) satisfying x^2 - 13xy + y^2 + 11 = 0. - Colin Barker, Feb 10 2014

It appears that the b-file, formulas and programs are based on the conjectured, so far apparently unproved recurrence relation. - M. F. Hasler, Nov 05 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..900

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3 (2) (2008), pp. 76-114. See Section 13.

Tanya Khovanova, Recursive Sequences

John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types

John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types [local copy, corrected]

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Index entries for linear recurrences with constant coefficients, signature (13,-1).

Formula

It appears that the sequence satisfies a(n+1) = 13*a(n) - a(n-1). [Corrected by M. F. Hasler, Nov 05 2018]

If the recurrence a(n+2) = 13*a(n+1) - a(n) holds then for n > 0, a(n)*a(n+3) = 143 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004

G.f.: x*(1-x)/(1 - 13*x + x^2). - Philippe Deléham, Nov 17 2008

For n>1, a(n) is the numerator of the continued fraction [1,11,1,11,...,1,11] with (n-1) repetitions of 1,11. - Greg Dresden, Sep 10 2019

Mathematica

CoefficientList[Series[(1 - x)/(1 - 13 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{13, -1}, {1, 12}, 30] (* G. C. Greubel, Jan 18 2018 *)

Prog

(PARI) x='x+O('x^30); Vec(x*(1-x)/(1-13*x+x^2)) \\ G. C. Greubel, Jan 18 2018
(Magma) I:=[1, 12]; [n le 2 select I[n] else 13*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

Crossrefs

Cf. A078362, A082630, A082981.

Row 13 of array A094954.

Cf. similar sequences listed in A238379.

Sequence in context: A036360 A120657 A015612 * A082173 A005723 A097259

Adjacent sequences: A085257 A085258 A085259 * A085261 A085262 A085263

Keyword

nonn

Author

John W. Layman, Jun 23 2003

Status

approved