A168520 -id:A168520 - cp's OEIS frontend

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

N. J. A. Sloane

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)

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).

Cf. A099374 (squares).
Cf. A041040.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A243399.
Row n=10 of A073133, A172236 and A352361.

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

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

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

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

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

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

Extended by T. D. Noe, May 23 2011

A168522 a(n) = 98a(n-1) - 2a(n-2); a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 98, 9602, 940800, 92179196, 9031679608, 884920243192, 86704120473600, 8495233965926416, 832359520419841568, 81554242533212620832, 7990651049213997158400, 782920694337905296281536, 76710246743016291041273728, 7516038339426920711452262272

Offset: 1

Mark Dols, Nov 28 2009

a(n)/a(n+1) converges to 49 - sqrt(2399). - corrected by Mark Dols, Jun 20 2010

G. C. Greubel, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (98,-2).

Cf. A168520, A004189, A000108, A000984, A165293

[n le 2 select n-1 else 98*Self(n-1)-2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jul 25 2016

LinearRecurrence[{98,-2},{0,1}, 25] (* G. C. Greubel, Jul 25 2016 *)

G.f.: x^2/(1 - 98x + 2x^2). - Philippe Deléham, Nov 29 2009

E.g.f.: (1/(2*b))*exp(49*x)*( 49*sinh(b*x) - b*cosh(b*x) ) - (1/2), where b = sqrt(2399). - G. C. Greubel, Jul 25 2016

Definition adapted to offset by Georg Fischer, Jun 19 2021

A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

Original entry on oeis.org

7, 799, 78407, 7683199, 752875207, 73774087199, 7229107670407, 708378777612799, 69413891098384007, 6801852948864019999, 666512175097575576007, 65311391306613542428799, 6399849835873029582446407, 627119972524250285537319199, 61451357457540654953074835207

Offset: 1

Bruno Berselli, Nov 23 2016

It is well known that T(m) + k*T(m+1) is always a square for k=1. For k=3, the nonnegative values of m are the terms of A278310.
Square roots of T(m) + 2*T(m+1) are listed by A168520 (after 0).
Negative values of m for which T(m) + 2*T(m+1) is a square: -1, -2, -82, -7922, -776162, ...

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Subsequence of A056220.
Cf. A000217, A001078, A001079, A122652, A168520, A278620.
Cf. A278310: numbers m such that T(m) + 3*T(m+1) is a square.

Iv:=[7, 799]; [n le 2 select Iv[n] else 98*Self(n-1)-Self(n-2)+112: n in [1..20]];

P:=proc(q) local n; for n from 1 to q do if type(sqrt((3*n^2+7*n+4)/2),integer) then print(n); fi; od; end: P(10^9); #  Paolo P. Lava, Nov 25 2016

Table[((5 + 2 Sqrt[6])^(2 n) + (5 - 2 Sqrt[6])^(2 n))/12 - 7/6, {n, 1, 20}]
RecurrenceTable[{a[1] == 7, a[2] == 799, a[n] == 98 a[n - 1] - a[n - 2] + 112}, a, {n, 1, 20}]
LinearRecurrence[{99,-99,1},{7,799,78407},20] (* Harvey P. Dale, Oct 18 2024 *)

Vec(x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)) + O(x^20)) \\ Colin Barker, Nov 27 2016

def A278438():
    a, b = 7, 799
    yield a
    while True:
        yield b
        a, b = b, 98*b - a + 112
a = A278438(); print([next(a) for  in range(15)]) # _Peter Luschny, Nov 24 2016

O.g.f.: x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)).
E.g.f.: (exp((5-2*sqrt(6))^2*x) + exp((5+2*sqrt(6))^2*x) - 14*exp(x))/12 + 1.
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>3.
a(n) = 98*a(n-1) - a(n-2) + 112 for n>2.
a(n) = a(-n) = ((5 + 2*sqrt(6))^(2*n) + (5 - 2*sqrt(6))^(2*n))/12 - 7/6.
a(n) = A001079(2*n)/6 - 7/6.
a(n) = 2*A001078(n)^2 - 1 = A122652(n)^2/2 - 1.
a(n) = -A278620(n+1) + 106*A278620(n) + 7*A278620(n-1).
Lim_{n -> infinity} a(n)/a(n-1) = (5 + 2*sqrt(6))^2.

A171415 a(n) = 99*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 99, 9800, 970101, 96030199, 9506019600, 940999910201, 93149485090299, 9220858024029400, 912771794893820301, 90355186836464180399, 8944250725015060039200, 885390466589654479700401, 87644711941650778430300499, 8675941091756837410120049000, 858830523371985252823454550501

Offset: 0

Mark Dols, Dec 08 2009

Related to Motzkin numbers.

Index entries for linear recurrences with constant coefficients, signature (99,-1).

Cf. A168520, A168522, A004189.

a(0):=0: a(1):=1: for n from 0 to 50 do a(n+2):=99*a(n+1)-a(n): od: seq(a(n),n=0..30);
taylor((z/(1-99*z+z^2)),z=0,30); # Richard Choulet, Dec 10 2009

LinearRecurrence[{99,-1},{0,1},30] (* Harvey P. Dale, Dec 18 2015 *)

a(n+1)^2 - a(n)^2 = a(2*n+1). - Richard Choulet, Dec 10 2009
G.f.: x/(1-99*x+x^2). - Philippe Deléham, Dec 09 2009
E.g.f.: 2*exp(99*x/2)*sinh(sqrt(9797)*x/2)/sqrt(9797). - Stefano Spezia, Aug 05 2024

Offset adapted to definition by Georg Fischer, Jun 18 2021

a(14)-a(16) from Stefano Spezia, Aug 05 2024

cp's OEIS Frontend

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

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A168522 a(n) = 98*a(n-1) - 2*a(n-2); a(1) = 0, a(2) = 1.

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A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

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Magma

Maple

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A171415 a(n) = 99*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.

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Maple

Mathematica

Formula

Extensions

A168522 a(n) = 98a(n-1) - 2a(n-2); a(1) = 0, a(2) = 1.