A041151 -id:A041151 - cp's OEIS frontend

A099371 Expansion of g.f.: x/(1 - 9*x - x^2).

0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, 426938895, 3889316089, 35430783696, 322766369353, 2940328107873, 26785719340210, 244011802169763, 2222891938868077, 20250039251982456, 184473245206710181, 1680509246112374085, 15309056460218076946

Offset: 0

Wolfdieter Lang, Oct 18 2004

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 9's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
For n >= 1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,9} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Mar 10 2023: (Start)
Also called the 9-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 9 kinds of squares available. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
J. H. Han and M. D. Hirschhorn, Another Look at an Amazing Identity of Ramanujan, Mathematics Magazine, Vol. 79 (2006), pp. 302-304. See equation 6 on page 303.
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 (9,1).

Row n=9 of A073133, A172236 and A352361.
Cf. A099372 (squares).
Cf. A041151, A054413, A086902, A087798, A099371, A097839, A178765, A243399.

I:=[0,1]; [n le 2 select I[n] else 9*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

F:= gfun:-rectoproc({a(n)=9*a(n-1)+a(n-2),a(0)=0,a(1)=1},a(n),remember):
seq(F(n),n=0..30); # Robert Israel, Feb 01 2015

CoefficientList[Series[x/(1-9*x-x^2), {x,0,30}], x] (* G. C. Greubel, Apr 16 2017 *)
LinearRecurrence[{9,1}, {0,1}, 30] (* G. C. Greubel, Jan 24 2018 *)

my(x='x+O('x^30)); concat([0], Vec(1/(1-9*x-x^2)) ) \\ Charles R Greathouse IV, Feb 03 2014

from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(0,1,9,9,1,0)
[next(it) for i in range(1,22)] # Zerinvary Lajos, Jul 09 2008

[lucas_number1(n,9,-1) for n in range(0, 20)] # Zerinvary Lajos, Apr 26 2009

G.f.: x/(1 - 9*x - x^2).
a(n) = 9*a(n-1) + a(n-2), n >= 2, a(0)=0, a(1)=1.
a(n) = (-i)^(n-1)*S(n-1, 9*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.
a(n) = (ap^n - am^p)/(ap-am) with ap:= (9+sqrt(85))/2 and am:= (9-sqrt(85))/2 = -1/ap (Binet form).
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1-k, k)*9^(n-1-2*k) n >= 1.
a(n) = F(n, 9), the n-th Fibonacci polynomial evaluated at x=9. - T. D. Noe, Jan 19 2006
a(n) = ((9+sqrt(85))^n - (9-sqrt(85))^n)/(2^n*sqrt(85)). Offset 1. a(3)=82. - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
a(p) == 85^((p-1)/2) (mod p) for odd primes p. - Gary W. Adamson, Feb 22 2009 [See A087475 for more info about this congruence. - Jason Yuen, Apr 05 2025]
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+2) = 9*A097839(n), a(2n+1) = A097841(n).
a(3n+1) = A041151(5n), a(3n+2) = A041151(5n+3), a(3n+3) = 2*A041151(5n+4).
Limit_{k -> infinity} (a(n+k)/a(k)) = (A087798(n) + A099371(n)*sqrt(85))/2.
Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)
a(n) ~ 1/sqrt(85)*((9+sqrt(85))/2)^n. - Jean-François Alcover, Dec 04 2013
a(n) = [1,0] (M^n) [0,1]^T where M is the matrix [9,1; 1,0]. - Robert Israel, Feb 01 2015
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(85)*x/2)/sqrt(85). - Stefano Spezia, Apr 06 2023

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

A041150 Numerators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

9, 37, 46, 83, 378, 6887, 27926, 34813, 62739, 285769, 5206581, 21112093, 26318674, 47430767, 216041742, 3936182123, 15960770234, 19896952357, 35857722591, 163327842721, 2975758891569, 12066363408997

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 17 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A087798.
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. A010536, A041018, A041046, A041090, A041150, A041151, A041226, A041318, A041426,

A041550.

Numerator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

From Johannes W. Meijer, Jun 17 2010: (Start)
a(5*n) = A087798(3*n+1), a(5*n+1) = (A087798(3*n+2) - A087798(3*n+1))/2, a(5*n+2) = (A087798(3*n+2) + A087798(3*n+1))/2, a(5*n+3) = A087798(3*n+2) and a(5*n+4) = A087798(3*n+3)/2. (End)
G.f.: -(x^9-9*x^8+37*x^7-46*x^6+83*x^5+378*x^4+83*x^3+46*x^2+37*x+9) / (x^10+756*x^5-1). - Colin Barker, Nov 04 2013

A041227 Denominators of continued fraction convergents to sqrt(125).

Original entry on oeis.org

1, 5, 6, 11, 61, 1353, 6826, 8179, 15005, 83204, 1845493, 9310669, 11156162, 20466831, 113490317, 2517253805, 12699759342, 15217013147, 27916772489, 154800875592, 3433536035513, 17322481053157, 20756017088670, 38078498141827, 211148507797805

Offset: 0

N. J. A. Sloane

The a(n) terms of this sequence can be constructed with the terms of sequence A049666. For the terms of the periodical sequence of the continued fraction for sqrt(125) see A010186. 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 (0,0,0,0,1364,0,0,0,0,1).

Cf. A041226, A010186, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

I:=[1, 5, 6, 11, 61, 1353, 6826, 8179, 15005, 83204]; [n le 10 select I[n] else 1364*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 13 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[125], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[125], 30]]  (* Vincenzo Librandi, Dec 13 2013 *)
LinearRecurrence[{0,0,0,0,1364,0,0,0,0,1},{1,5,6,11,61,1353,6826,8179,15005,83204},30] (* Harvey P. Dale, Apr 29 2022 *)

a(5*n) = A049666(3*n+1), a(5*n+1) = (A049666(3*n+2) - A049666(3*n+1))/2, a(5*n+2) = (A049666(3*n+2)+A049666(3*n+1))/2, a(5*n+3):= A049666(3*n+2) and a(5*n+4) = A049666(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: -(x^8 -5*x^7 +6*x^6 -11*x^5 +61*x^4 +11*x^3 +6*x^2 +5*x +1) / ((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 12 2013
a(n) = 1364*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 13 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

A099371 Expansion of g.f.: x/(1 - 9*x - x^2).

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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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A041150 Numerators of continued fraction convergents to sqrt(85).

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A041227 Denominators of continued fraction convergents to sqrt(125).

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

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