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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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).

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.

Programs

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

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 *)

PARI

x='x+O('x^30); Vec(1/(1-10*x-x^2)) \\ G. C. Greubel, Jan 24 2018

Sage

[lucas_number1(n,10,-1) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009

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

Extensions

Extended by T. D. Noe, May 23 2011