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

1, 10, 101, 1020, 10301, 104030, 1050601, 10610040, 107151001, 1082120050, 10928351501, 110365635060, 1114584702101, 11256212656070, 113676711262801, 1148023325284080, 11593909964103601, 117087122966320090, 1182465139627304501, 11941738519239365100

Offset

0,2

Comments

Generalized Fibonacci sequence.

Sqrt(26) = 10/2 + 10/101 + 10/(101*10301) + 10/(10301*1050601) + ... - Gary W. Adamson, Jun 13 2008

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

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

From Bruno Berselli, May 03 2018: (Start)

Numbers k for which m*k^2 + (-1)^k is a perfect square:

m = 2: 0, 1, 2, 5, 12, 29, 70, 169, ... (A000129);

m = 3: 0, 4, 56, 780, 10864, 151316, ... (4*A007655);

m = 5: 0, 1, 4, 17, 72, 305, 1292, ... (A001076);

m = 6: 0, 2, 20, 198, 1960, 19402, ... (A001078);

m = 7: 0, 48, 12192, 3096720, ... (2*A175672);

m = 8: 0, 6, 204, 6930, 235416, ... (A082405);

m = 10: 0, 1, 6, 37, 228, 1405, 8658, ... (A005668);

m = 11: 0, 60, 23880, 9504180, ... [°];

m = 12: 0, 2, 28, 390, 5432, 75658, ... (A011944);

m = 13: 0, 5, 180, 6485, 233640, ... (5*A041613);

m = 14: 0, 4, 120, 3596, 107760, ... (A068204);

m = 15: 0, 8, 496, 30744, 1905632, ... [°];

m = 17: 0, 1, 8, 65, 528, 4289, 34840, ... (A041025);

m = 18: 0, 4, 136, 4620, 156944, ... (A202299);

m = 19: 0, 13260, 1532829480, ... [°];

m = 20: 0, 2, 36, 646, 11592, 208010, ... (A207832);

m = 21: 0, 12, 1320, 145188, ... (A174745);

m = 22: 0, 42, 16548, 6519870, ... (A174766);

m = 23: 0, 240, 552480, 1271808720, ... [°];

m = 24: 0, 10, 980, 96030, 9409960, ... (A168520);

m = 26: 0, 1, 10, 101, 1020, 10301, ... (this sequence);

m = 27: 0, 260, 702520, 1898208780, ... [°];

m = 28: 0, 24, 6096, 1548360, ... (A175672);

m = 29: 0, 13, 1820, 254813, 35675640, ... [°];

m = 30: 0, 2, 44, 966, 21208, 465610, ... (2*A077421), etc.

[°] apparently without related sequences in the OEIS.

(End)

From Michael A. Allen, Mar 12 2023: (Start)

Also called the 10-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

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)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.

Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.

Sergio Falcón and Ángel Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007).

Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (10,1).

Formula

G.f.: 1/(1 - 10*x - x^2).

a(n) = 10*a(n-1) + a(n-2), n>=1; a(-1):=0, a(0)=1.

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.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = 5+sqrt(26), am = -1/ap = 5-sqrt(26).

a(n) = F(n+1, 10), the (n+1)-th Fibonacci polynomial evaluated at x=10. - T. D. Noe, Jan 19 2006

a(n) = Sum_{i=0..floor(n/2)} binomial(n-i,i)*10^(n-2*i). - Sergio Falcon, Sep 24 2007

Maple

seq(combinat:-fibonacci(n+1, 10), n=0..19); # Peter Luschny, May 04 2018

Mathematica

Denominator[Convergents[Sqrt[26], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{10, 1}, {1, 10}, 30] (* G. C. Greubel, Jan 24 2018 *)

Prog

(Sage) [lucas_number1(n, 10, -1) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009
(PARI) x='x+O('x^30); Vec(1/(1-10*x-x^2)) \\ G. C. Greubel, Jan 24 2018
(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

Crossrefs

Cf. A099374 (squares).

Cf. A041040.

Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A243399.

Row n=10 of A073133, A172236 and A352361.

Sequence in context: A180175 A267526 A261199 * A333344 A163461 A081192

Adjacent sequences: A041038 A041039 A041040 * A041042 A041043 A041044

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

Extended by T. D. Noe, May 23 2011

Status

approved