A041227 -id:A041227 - cp's OEIS frontend

A049666 a(n) = Fibonacci(5*n)/5.

0, 1, 11, 122, 1353, 15005, 166408, 1845493, 20466831, 226980634, 2517253805, 27916772489, 309601751184, 3433536035513, 38078498141827, 422297015595610, 4683345669693537, 51939099382224517, 576013438874163224

Offset: 0

Clark Kimberling

For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 11's along the main diagonal and 1's along the subdiagonal and the superdiagonal. - John M. Campbell, Jul 08 2011
For n >= 1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,11} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
For n >= 1, a(n) equals the denominator of the continued fraction [11, 11, ..., 11] (with n copies of 11). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 26 2019
From Michael A. Allen, Mar 30 2023: (Start)
Also called the 11-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 11 kinds of squares available. (End)

			G.f. = x + 11*x^2 + 122*x^3 + 1353*x^4 + 15005*x^5 + 166408*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..950
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (11,1).

A column of array A028412.
Cf. A000045, A102312, A243399.
Row n=11 of A073133, A172236 and A352361, and column k=11 of A157103.

[Fibonacci(5*n)/5: n in [0..30]]; // G. C. Greubel, Dec 02 2017

A049666 := proc(n)
    combinat[fibonacci](5*n)/5 ;
end proc: # R. J. Mathar, May 07 2024

Table[Fibonacci[5*n]/5, {n, 0, 100}] (* T. D. Noe, Oct 29 2009 *)
a[ n_] := Fibonacci[n, 11]; (* Michael Somos, May 28 2014 *)

numlib::fibonacci(5*n)/5 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(5*n)/5 \\ Charles R Greathouse IV, Feb 03 2014

from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(0,1,11,11,1,0)
[next(it) for i in range(1,22)] # Zerinvary Lajos, Jul 09 2008

[lucas_number1(n,11,-1) for n in range(0, 19)] # Zerinvary Lajos, Apr 27 2009

[fibonacci(5*n)/5 for n in range(0, 19)] # Zerinvary Lajos, May 15 2009

G.f.: x/(1 - 11*x - x^2).
a(n) = A102312(n)/5.
a(n) = 11*a(n-1) + a(n-2) for n > 1, a(0)=0, a(1)=1. With a=golden ratio and b=1-a, a(n) = (a^(5n)-b^(5n))/(5*sqrt(5)). - Mario Catalani (mario.catalani(AT)unito.it), Jul 24 2003
a(n) = F(n, 11), the n-th Fibonacci polynomial evaluated at x=11. - T. D. Noe, Jan 19 2006
a(n) = ((11+sqrt(125))^n-(11-sqrt(125))^n)/(2^n*sqrt(125)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n) = 11*A049670(n), a(2n+1) = A097843(n).
a(3n+1) = A041227(5n), a(3n+2) = A041227(5n+3), a(3n+3) = 2*A041227(5n+4).
Limit_{k->oo} a(n+k)/a(k) = (A001946(n) + A049666(n)*sqrt(125))/2.
Limit_{n->oo} A001946(n)/A049666(n) = sqrt(125).
(End)
a(n) = F(n) + (-1)^n*5*F(n)^3 + 5*F(n)^5, n >= 0. See the D. Jennings formula given in a comment on A111125, where also the reference is given. - Wolfdieter Lang, Aug 31 2012
a(-n) = -(-1)^n * a(n). - Michael Somos, May 28 2014
E.g.f.: (exp((1/2)*(11-5*sqrt(5))*x)*(-1 + exp(5*sqrt(5)*x)))/(5*sqrt(5)). - Stefano Spezia, Aug 02 2019

A041019 Denominators of continued fraction convergents to sqrt(13).

Original entry on oeis.org

1, 1, 2, 3, 5, 33, 38, 71, 109, 180, 1189, 1369, 2558, 3927, 6485, 42837, 49322, 92159, 141481, 233640, 1543321, 1776961, 3320282, 5097243, 8417525, 55602393, 64019918, 119622311, 183642229, 303264540, 2003229469, 2306494009, 4309723478, 6616217487, 10925940965

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,36,0,0,0,0,1).

Cf. A010122 (continued fraction for sqrt(13)), A041018 (numerators).

Cf. A041047, A041091, A041151, A041227, A041319, A041427 and A041551. - Johannes W. Meijer, Jun 12 2010

I:=[1, 1, 2, 3, 5, 33, 38, 71, 109, 180]; [n le 10 select I[n] else 36*Self(n-5)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[13], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
CoefficientList[Series[((1 - 2 x + 4 x^2 - 3 x^3 + x^4) (1 + 3 x + 4 x^2 + 2 x^3 + x^4))/(1 - 36 x^5 - x^10), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{0,0,0,0,36,0,0,0,0,1},{1,1,2,3,5,33,38,71,109,180},40] (* Harvey P. Dale, Sep 30 2016 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5*n) = A006190(3*n+1),
a(5*n+1) = (A006190(3*n+2) - A006190(3*n+1))/2,
a(5*n+2) = (A006190(3*n+2) + A006190(3*n+1))/2,
a(5*n+3) = A006190(3*n+2) and a(5*n+4) = A006190(3*n+3)/2. (End)
G.f.: ((1 - 2*x + 4*x^2 - 3*x^3 + x^4)*(1 + 3*x + 4*x^2 + 2*x^3 + x^4))/(1 - 36*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013
a(n) = A010122(n)*a(n-1) + a(n-2), a(0)=1, a(-1)=0. - Paul Weisenhorn, Aug 17 2018

More terms from Vincenzo Librandi, Dec 10 2013

A041047 Denominators of continued fraction convergents to sqrt(29).

Original entry on oeis.org

1, 2, 3, 5, 13, 135, 283, 418, 701, 1820, 18901, 39622, 58523, 98145, 254813, 2646275, 5547363, 8193638, 13741001, 35675640, 370497401, 776670442, 1147167843, 1923838285, 4994844413, 51872282415

Offset: 0

N. J. A. Sloane

The terms of this sequence can be constructed with the terms of sequence A052918.

For the terms of the periodical sequence of the continued fraction for sqrt(29) see A010128. We observe that its period is five. The decimal expansion of sqrt(29) is A010484. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,140,0,0,0,0,1).

Cf. A041046.

Cf. A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

I:=[1, 2, 3, 5, 13, 135, 283, 418, 701, 1820]; [n le 10 select I[n] else 140*Self(n-5)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[29],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[29], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

a(5*n) = A052918(3*n), a(5*n+1) = (A052918(3*n+1) - A052918(3*n))/2, a(5*n+2) = (A052918(3*n+1) + A052918(3*n))/2, a(5*n+3) = A052918(3*n+1) and a(5*n+4) = A052918(3*n+2)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: (1 + 2*x + 3*x^2 + 5*x^3 + 13*x^4 - 5*x^5 + 3*x^6 - 2*x^7 + x^8)/(1 - 140*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013
a(n) = 140*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 10 2013

A041091 Denominators of continued fraction convergents to sqrt(53).

Original entry on oeis.org

1, 3, 4, 7, 25, 357, 1096, 1453, 2549, 9100, 129949, 398947, 528896, 927843, 3312425, 47301793, 145217804, 192519597, 337737401, 1205731800, 17217982601, 52859679603, 70077662204, 122937341807, 438889687625, 6267392968557, 19241068593296, 25508461561853

Offset: 0

N. J. A. Sloane

The terms of this sequence can be constructed with the terms of sequence A054413. For the terms of the periodic sequence of the continued fraction for sqrt(53) see A010139. We observe that its period is five. The decimal expansion of sqrt(53) is A010506. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,364,0,0,0,0,1).

Cf. A010506, A041090.

Cf. A041019, A041047, A041151, A041227, A041319, A041427 and A041551. - Johannes W. Meijer, Jun 12 2010

convert(sqrt(53), confrac, 30, cvgts): denom(cvgts); # Wesley Ivan Hurt, Dec 17 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[53], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[53], 30]] (* Vincenzo Librandi, Oct 24 2013 *)
LinearRecurrence[{0,0,0,0,364,0,0,0,0,1},{1,3,4,7,25,357,1096,1453,2549,9100},30] (* Harvey P. Dale, Nov 13 2019 *)

a(5*n) = A054413(3*n), a(5*n+1) = (A054413(3*n+1) - A054413(3*n))/2, a(5*n+2)= (A054413(3*n+1) + A054413(3*n))/2, a(5*n+3) = A054413(3*n+1) and a(5*n+4) = A054413(3*n+2)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: -(x^8-3*x^7+4*x^6-7*x^5+25*x^4+7*x^3+4*x^2+3*x+1) / (x^10+364*x^5-1). - Colin Barker, Sep 26 2013

A041151 Denominators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996, 564733, 2289928, 2854661, 5144589, 23433017, 426938895, 1731188597, 2158127492, 3889316089, 17715391848, 322766369353, 1308780869260, 1631547238613, 2940328107873, 13392859670105, 244011802169763, 989440068349157

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A099371.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,756,0,0,0,0,1).

Cf. A041150, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

I:=[1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996]; [n le 10 select I[n] else 756*Self(n-5)+Self(n-10): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[85], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5*n) = A099371(3*n+1), a(5*n+1) = (A099371(3*n+2)-A099371(3*n+1))/2, a(5*n+2) = (A099371(3*n+2)+A099371(3*n+1))/2, a(5*n+3):= A099371(3*n+2) and a(5*n+4) = A099371(3*n+3)/2. (End)
G.f.: -(x^8-4*x^7+5*x^6-9*x^5+41*x^4+9*x^3+5*x^2+4*x+1) / (x^10+756*x^5-1). - Colin Barker, Nov 11 2013
a(n) = 756*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 12 2013

A041226 Numerators of continued fraction convergents to sqrt(125).

Original entry on oeis.org

11, 56, 67, 123, 682, 15127, 76317, 91444, 167761, 930249, 20633239, 104096444, 124729683, 228826127, 1268860318, 28143753123, 141987625933, 170131379056, 312119004989, 1730726404001, 38388099893011, 193671225869056, 232059325762067, 425730551631123

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A001946.
For the terms of the periodical sequence of the continued fraction for sqrt(125) see A010186. We observe that its period is five. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1364,0,0,0,0,1).

Cf. A041227, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Numerator[Convergents[Sqrt[125], 30]] (* Vincenzo Librandi, Oct 31 2013 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5n)   = A001946(3n+1),
a(5n+1) = (A001946(3n+2) - A001946(3n+1))/2,
a(5n+2) = (A001946(3n+2) + A001946(3n+1))/2,
a(5n+3) = A001946(3n+2),
a(5n+4) = A001946(3n+3)/2. (End)
G.f.: -(x^9 -11*x^8 +56*x^7 -67*x^6 +123*x^5 +682*x^4 +123*x^3 +67*x^2 +56*x +11) / ((x^2 +4*x -1)*(x^4 -7*x^3 +19*x^2 -3*x +1)*(x^4 +3*x^3 +19*x^2 +7*x +1)). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A041319 Denominators of continued fraction convergents to sqrt(173).

Original entry on oeis.org

1, 6, 7, 13, 85, 2223, 13423, 15646, 29069, 190060, 4970629, 30013834, 34984463, 64998297, 424974245, 11114328667, 67110946247, 78225274914, 145336221161, 950242601880, 24851643870041, 150060105822126, 174911749692167, 324971855514293, 2124742882777925

Offset: 0

N. J. A. Sloane

The a(n) terms of this sequence can be constructed with the terms of sequence A140455. For the terms of the periodical sequence of the continued fraction for sqrt(173) see A010217. We observe that its period is five. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 2236, 0, 0, 0, 0, 1).

Cf. A041318, A010217, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.

I:=[1,6,7,13,85,2223,13423,15646,29069,190060]; [n le 10 select I[n] else 2236*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 15 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[173], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[173], 30]] (* Vincenzo Librandi, Dec 15 2013 *)
LinearRecurrence[{0,0,0,0,2236,0,0,0,0,1},{1,6,7,13,85,2223,13423,15646,29069,190060},30] (* Harvey P. Dale, Sep 19 2020 *)

a(5*n) = A140455(3*n+1), a(5*n+1) = (A140455(3*n+2) - A140455(3*n+1))/2, a(5*n+2) = (A140455(3*n+2)+A140455(3*n+1))/2, a(5*n+3) = A140455(3*n+2) and a(5*n+4) = A140455(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: -(x^8-6*x^7+7*x^6-13*x^5+85*x^4+13*x^3+7*x^2+6*x+1) / (x^10+2236*x^5-1). - Colin Barker, Nov 12 2013
a(n) = 2236*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 15 2013

A041427 Denominators of continued fraction convergents to sqrt(229).

Original entry on oeis.org

1, 7, 8, 15, 113, 3405, 23948, 27353, 51301, 386460, 11645101, 81902167, 93547268, 175449435, 1321693313, 39826248825, 280105435088, 319931683913, 600037119001, 4520191516920, 136205782626601, 957960669903127, 1094166452529728, 2052127122432855

Offset: 0

N. J. A. Sloane

The a(n) terms of this sequence can be constructed with the terms of sequence A154597. For the terms of the periodical sequence of the continued fraction for sqrt(229) see A040213. We observe that its period is five. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 3420, 0, 0, 0, 0, 1).

Cf. A041426, A166125, A040213, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.

I:=[1,7,8,15,113,3405,23948,27353,51301,386460]; [n le 10 select I[n] else 3420*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 17 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[229], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[229], 30]] (* Vincenzo Librandi, Dec 17 2013 *)
LinearRecurrence[{0,0,0,0,3420,0,0,0,0,1},{1,7,8,15,113,3405,23948,27353,51301,386460},30] (* Harvey P. Dale, Oct 14 2020 *)

a(5*n) = A154597(3*n+1), a(5*n+1) = (A154597(3*n+2) - A154597(3*n+1))/2, a(5*n+2) = (A154597(3*n+2) + A154597(3*n+1))/2, a(5*n+3) = A154597(3*n+2) and a(5*n+4) = A154597(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: -(x^8 -7*x^7 +8*x^6 -15*x^5 +113*x^4 +15*x^3 +8*x^2 +7*x +1) / (x^10 +3420*x^5 -1). - Colin Barker, Nov 12 2013
a(n) = 3420*a(n-5) + a(n-10) for n>9. - Vincenzo Librandi, Dec 17 2013

A041551 Denominators of continued fraction convergents to sqrt(293).

Original entry on oeis.org

1, 8, 9, 17, 145, 4947, 39721, 44668, 84389, 719780, 24556909, 197175052, 221731961, 418907013, 3572988065, 121900501223, 978776997849, 1100677499072, 2079454496921, 17736313474440, 605114112627881, 4858649214497488, 5463763327125369, 10322412541622857

Offset: 0

N. J. A. Sloane

The a(n) terms of this sequence can be constructed with the terms of sequence A178765. For the terms of the periodical sequence of the continued fraction for sqrt(293) see A040275. We observe that its period is five. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 4964, 0, 0, 0, 0, 1).

Cf. A041550, A040275, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.

I:=[1,8,9,17,145,4947,39721,44668,84389,719780]; [n le 10 select I[n] else 4964*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 20 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[293], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[293 ], 30]] (* Vincenzo Librandi, Dec 20 2013 *)

a(5n) = A178765(3n), a(5n+1) = (A178765(3n+1) - A178765(3n))/2, a(5n+2) = (A178765(3n+1) + A178765(3n))/2, a(5n+3) = A178765(3n+1) and a(5n+4) = A178765(3n+2)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: -(x^8-8*x^7+9*x^6-17*x^5+145*x^4+17*x^3+9*x^2+8*x+1) / (x^10+4964*x^5-1). - Colin Barker, Nov 12 2013
a(n) = 4964*a(n-5) + a(n-10) for n>9. - Vincenzo Librandi, Dec 20 2013

cp's OEIS Frontend

A049666 a(n) = Fibonacci(5*n)/5.

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A041019 Denominators of continued fraction convergents to sqrt(13).

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A041047 Denominators of continued fraction convergents to sqrt(29).

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A041091 Denominators of continued fraction convergents to sqrt(53).

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A041151 Denominators of continued fraction convergents to sqrt(85).

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A041226 Numerators of continued fraction convergents to sqrt(125).

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A041319 Denominators of continued fraction convergents to sqrt(173).

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