cp's OEIS Frontend

A041041 Denominators of continued fraction convergents to sqrt(26).

%I A041041 #85 Jan 05 2025 19:51:35
%S A041041 1,10,101,1020,10301,104030,1050601,10610040,107151001,1082120050,
%T A041041 10928351501,110365635060,1114584702101,11256212656070,
%U A041041 113676711262801,1148023325284080,11593909964103601,117087122966320090,1182465139627304501,11941738519239365100
%N A041041 Denominators of continued fraction convergents to sqrt(26).
%C A041041 Generalized Fibonacci sequence.
%C A041041 Sqrt(26) = 10/2 + 10/101 + 10/(101*10301) + 10/(10301*1050601) + ... - _Gary W. Adamson_, Jun 13 2008
%C A041041 For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 10's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C A041041 a(n) equals the number of words of length n on alphabet {0, 1, ..., 10} avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015
%C A041041 From _Bruno Berselli_, May 03 2018: (Start)
%C A041041 Numbers k for which m*k^2 + (-1)^k is a perfect square:
%C A041041 m =  2: 0, 1, 2, 5, 12, 29, 70, 169, ...   (A000129);
%C A041041 m =  3: 0, 4, 56, 780, 10864, 151316, ...  (4*A007655);
%C A041041 m =  5: 0, 1, 4, 17, 72, 305, 1292, ...    (A001076);
%C A041041 m =  6: 0, 2, 20, 198, 1960, 19402, ...    (A001078);
%C A041041 m =  7: 0, 48, 12192, 3096720, ...         (2*A175672);
%C A041041 m =  8: 0, 6, 204, 6930, 235416, ...       (A082405);
%C A041041 m = 10: 0, 1, 6, 37, 228, 1405, 8658, ...  (A005668);
%C A041041 m = 11: 0, 60, 23880, 9504180, ...         [°];
%C A041041 m = 12: 0, 2, 28, 390, 5432, 75658, ...    (A011944);
%C A041041 m = 13: 0, 5, 180, 6485, 233640, ...       (5*A041613);
%C A041041 m = 14: 0, 4, 120, 3596, 107760, ...       (A068204);
%C A041041 m = 15: 0, 8, 496, 30744, 1905632, ...     [°];
%C A041041 m = 17: 0, 1, 8, 65, 528, 4289, 34840, ... (A041025);
%C A041041 m = 18: 0, 4, 136, 4620, 156944, ...       (A202299);
%C A041041 m = 19: 0, 13260, 1532829480, ...          [°];
%C A041041 m = 20: 0, 2, 36, 646, 11592, 208010, ...  (A207832);
%C A041041 m = 21: 0, 12, 1320, 145188, ...           (A174745);
%C A041041 m = 22: 0, 42, 16548, 6519870, ...         (A174766);
%C A041041 m = 23: 0, 240, 552480, 1271808720, ...    [°];
%C A041041 m = 24: 0, 10, 980, 96030, 9409960, ...    (A168520);
%C A041041 m = 26: 0, 1, 10, 101, 1020, 10301, ...    (this sequence);
%C A041041 m = 27: 0, 260, 702520, 1898208780, ...    [°];
%C A041041 m = 28: 0, 24, 6096, 1548360, ...          (A175672);
%C A041041 m = 29: 0, 13, 1820, 254813, 35675640, ... [°];
%C A041041 m = 30: 0, 2, 44, 966, 21208, 465610, ...  (2*A077421), etc.
%C A041041 [°] apparently without related sequences in the OEIS.
%C A041041 (End)
%C A041041 From _Michael A. Allen_, Mar 12 2023: (Start)
%C A041041 Also called the 10-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C A041041 a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 10 kinds of squares available. (End)
%H A041041 Vincenzo Librandi, <a href="/A041041/b041041.txt">Table of n, a(n) for n = 0..200</a>
%H A041041 Michael A. Allen and Kenneth Edwards, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H A041041 Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.09.022">On the Fibonacci k-numbers</a>, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
%H A041041 Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.10.022">The k-Fibonacci sequence and the Pascal 2-triangle</a> Chaos, Solitons & Fractals 2007; 33(1): 38-49.
%H A041041 Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2007.03.007">On k-Fibonacci sequences and polynomials and their derivatives</a>, Chaos, Solitons & Fractals (2007).
%H A041041 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A041041 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A041041 Kai Wang, <a href="https://www.researchgate.net/publication/339487198_On_k-Fibonacci_Sequences_And_Infinite_Series_List_of_Results_and_Examples">On k-Fibonacci Sequences And Infinite Series List of Results and Examples</a>, 2020.
%H A041041 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A041041 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,1).
%F A041041 G.f.: 1/(1 - 10*x - x^2).
%F A041041 a(n) = 10*a(n-1) + a(n-2), n>=1; a(-1):=0, a(0)=1.
%F A041041 a(n) = S(n, 10*i)*(-i)^n where i^2:=-1 and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.
%F A041041 a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = 5+sqrt(26), am = -1/ap = 5-sqrt(26).
%F A041041 a(n) = F(n+1, 10), the (n+1)-th Fibonacci polynomial evaluated at x=10. - _T. D. Noe_, Jan 19 2006
%F A041041 a(n) = Sum_{i=0..floor(n/2)} binomial(n-i,i)*10^(n-2*i). - _Sergio Falcon_, Sep 24 2007
%p A041041 seq(combinat:-fibonacci(n+1, 10), n=0..19); # _Peter Luschny_, May 04 2018
%t A041041 Denominator[Convergents[Sqrt[26], 30]] (* _Vincenzo Librandi_, Dec 10 2013 *)
%t A041041 LinearRecurrence[{10,1}, {1,10}, 30] (* _G. C. Greubel_, Jan 24 2018 *)
%o A041041 (Sage) [lucas_number1(n,10,-1) for n in range(1, 19)] # _Zerinvary Lajos_, Apr 26 2009
%o A041041 (PARI) x='x+O('x^30); Vec(1/(1-10*x-x^2)) \\ _G. C. Greubel_, Jan 24 2018
%o A041041 (Magma) I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1) + Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 24 2018
%Y A041041 Cf. A099374 (squares).
%Y A041041 Cf. A041040.
%Y A041041 Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A243399.
%Y A041041 Row n=10 of A073133, A172236 and A352361.
%K A041041 nonn,frac,easy
%O A041041 0,2
%A A041041 _N. J. A. Sloane_
%E A041041 Extended by _T. D. Noe_, May 23 2011