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%I #69 Nov 02 2022 11:54:31
%S 1,12,155,2003,25884,334489,4322473,55857660,721827107,9327894731,
%T 120540804396,1557702562417,20129592507025,260127000028908,
%U 3361521407868779,43439651302265219,561353945521579068
%N Ratio-determined insertion sequence I(0.0833344) (see the link below).
%C 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.
%C a(n) = L(n,13), where L is defined as in A108299. - _Reinhard Zumkeller_, Jun 01 2005
%C 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
%C Seems to be positive values of x (or y) satisfying x^2 - 13xy + y^2 + 11 = 0. - _Colin Barker_, Feb 10 2014
%C 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
%H Vincenzo Librandi, <a href="/A085260/b085260.txt">Table of n, a(n) for n = 1..900</a>
%H A. Fink, R. K. Guy and M. Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contrib. Discr. Math. 3 (2) (2008), pp. 76-114. See Section 13.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H John W. Layman, <a href="https://personal.math.vt.edu/layman/sequences/ins_seq.htm">Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types</a>
%H John W. Layman, <a href="/A085376/a085376.txt">Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types</a> [local copy, corrected]
%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-1).
%F It appears that the sequence satisfies a(n+1) = 13*a(n) - a(n-1). [Corrected by _M. F. Hasler_, Nov 05 2018]
%F 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
%F G.f.: x*(1-x)/(1 - 13*x + x^2). - _Philippe Deléham_, Nov 17 2008
%F 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
%t CoefficientList[Series[(1 - x)/(1 - 13 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%t LinearRecurrence[{13,-1}, {1,12}, 30] (* _G. C. Greubel_, Jan 18 2018 *)
%o (PARI) x='x+O('x^30); Vec(x*(1-x)/(1-13*x+x^2)) \\ _G. C. Greubel_, Jan 18 2018
%o (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
%Y Cf. A078362, A082630, A082981.
%Y Row 13 of array A094954.
%Y Cf. similar sequences listed in A238379.
%K nonn
%O 1,2
%A _John W. Layman_, Jun 23 2003
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