

A085260


Ratiodetermined insertion sequence I(0.0833344) (see the link below).


9



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

1,2


COMMENTS

This sequence is the ratiodetermined 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 (2n2) X (2n2) 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 bfile, 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. 76114. See Section 13.
Tanya Khovanova, Recursive Sequences
John W. Layman, RatioDetermined Insertion Sequences and the Tree of their Recurrence Types
J.C. Novelli, J.Y. Thibon, Hopf Algebras of mpermutations,(m+1)ary trees, and mparking 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(n1). [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*(1x)/(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 (n1) 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*(1x)/(113*x+x^2)) \\ G. C. Greubel, Jan 18 2018
(MAGMA) I:=[1, 12]; [n le 2 select I[n] else 13*Self(n1)  Self(n2): 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



