A002530 -id:A002530 - cp's OEIS frontend

A041083 Denominators of continued fraction convergents to sqrt(48).

1, 1, 13, 14, 181, 195, 2521, 2716, 35113, 37829, 489061, 526890, 6811741, 7338631, 94875313, 102213944, 1321442641, 1423656585, 18405321661, 19828978246, 256353060613, 276182038859, 3570537526921, 3846719565780, 49731172316281, 53577891882061

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 12 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Cf. A010502, A041082, A002530.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[48], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[48], 40]] (* Vincenzo Librandi, Oct 24 2013 *)
LinearRecurrence[{0,14,0,-1},{1,1,13,14},30] (* Harvey P. Dale, Mar 15 2015 *)

From Colin Barker, Jul 15 2012: (Start)
a(n) = 14*a(n-2) - a(n-4).
G.f.: (1+x-x^2)/((1-4*x+x^2)*(1+4*x+x^2)). (End)
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(3) + 2 and beta = sqrt(3) - 2 be the roots of the equation x^2 - sqrt(12)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 12 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 12*a(2*n) + a(2*n - 1). (End)

A041219 Denominators of continued fraction convergents to sqrt(120).

Original entry on oeis.org

1, 1, 21, 22, 461, 483, 10121, 10604, 222201, 232805, 4878301, 5111106, 107100421, 112211527, 2351330961, 2463542488, 51622180721, 54085723209, 1133336644901, 1187422368110, 24881784007101, 26069206375211, 546265911511321, 572335117886532, 11992968269241961

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 20 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,22,0,-1).

Cf. A041218, A002530.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[120], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[120],30]] (* Harvey P. Dale, Mar 14 2013 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 22 x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 24 2013 *)

From Colin Barker, Jul 15 2012: (Start)
a(n) = 22*a(n-2) - a(n-4).
G.f.: (1+x-x^2)/(1-22*x^2+x^4). (End)
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(5) + sqrt(6) and beta = sqrt(5) - sqrt(6) be the roots of the equation x^2 - sqrt(20)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 20 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 20*a(2*n) + a(2*n - 1). (End)

A041611 Denominators of continued fraction convergents to sqrt(323).

Original entry on oeis.org

1, 1, 35, 36, 1259, 1295, 45289, 46584, 1629145, 1675729, 58603931, 60279660, 2108112371, 2168392031, 75833441425, 78001833456, 2727895778929, 2805897612385, 98128414600019, 100934312212404, 3529895029821755, 3630829342034159, 126978092658983161

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 34 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,36,0,-1).

Cf. A041610, A040305, A002530.

I:=[1,1,35,36]; [n le 4 select I[n] else 36*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 21 2013

Denominator[Convergents[Sqrt[323], 30]] (* Vincenzo Librandi Dec 21 2013 *)

G.f.: -(x^2-x-1) / (x^4-36*x^2+1). - Colin Barker, Nov 19 2013
a(n) = 36*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Dec 21 2013
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(34) + sqrt(38) )/2 and beta = ( sqrt(34) - sqrt(38) )/2 be the roots of the equation x^2 - sqrt(34)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 34 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 34*a(2*n) + a(2*n - 1). (End)

More terms from Colin Barker, Nov 19 2013

A042011 Denominators of continued fraction convergents to sqrt(528).

Original entry on oeis.org

1, 1, 45, 46, 2069, 2115, 95129, 97244, 4373865, 4471109, 201102661, 205573770, 9246348541, 9451922311, 425130930225, 434582852536, 19546776441809, 19981359294345, 898726585392989, 918707944687334, 41321876151635685, 42240584096323019, 1899907576389848521

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 44 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,46,0,-1).

Cf. A042010, A040505, A002530, A041241.

I:=[1,1,45,46]; [n le 4 select I[n] else 46*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 12 2014

Denominator[Convergents[Sqrt[528], 20]] (* Harvey P. Dale, Nov 14 2011 *)
CoefficientList[Series[(1 + x - x^2)/(x^4 - 46 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 12 2014 *)

G.f.: -(x^2-x-1) / (x^4-46*x^2+1). - Colin Barker, Nov 29 2013
a(n) = 46*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Jan 12 2014
From Peter Bala, May 27 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(11) + sqrt(12) and beta = sqrt(11) - sqrt(12) be the roots of the equation x^2 - sqrt(44)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 44 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 44*a(2*n) + a(2*n - 1). (End)
a(2*n) = A041241(2*n) = numerator of continued fraction [4,11,4,11,...,4,11] with n pairs of 4,11. - Greg Dresden, Aug 10 2021

More terms from Colin Barker, Nov 29 2013

A042859 Denominators of continued fraction convergents to sqrt(960).

Original entry on oeis.org

1, 1, 61, 62, 3781, 3843, 234361, 238204, 14526601, 14764805, 900414901, 915179706, 55811197261, 56726376967, 3459393815281, 3516120192248, 214426605350161, 217942725542409, 13290990137894701, 13508932863437110, 823826961944121301, 837335894807558411

Offset: 0

N. J. A. Sloane, Dec 11 1999

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 60 and Q = -1. This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 26 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,62,0,-1).

Cf. A042858, A040929.

Cf. A002530.

I:=[1,1,61,62]; [n le 4 select I[n] else 62*Self(n-2)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2013

Denominator[Convergents[Sqrt[960], 30]] (* Vincenzo Librandi, Dec 25 2013 *)

G.f.: -(x^2-x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - Colin Barker, Dec 25 2013
a(n) = 62*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 25 2013
From Peter Bala, May 26 2014: (Start)
  The following remarks assume an offset of 1:
  Let alpha = sqrt(15) + 4 and beta = sqrt(15) - 4 be the roots of the equation x^2 - sqrt(60)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
  a(n) = product {k = 1..floor((n-1)/2)} ( 60 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 60*a(2*n) + a(2*n - 1). (End)

More terms from Colin Barker, Dec 25 2013

A125077 #4 in an infinite set of generalized Pascal's triangles with trigonometric properties.

Original entry on oeis.org

1, 1, 3, 1, 4, 11, 1, 7, 15, 41, 1, 8, 38, 56, 153, 1, 11, 46, 186, 209, 571, 1, 12, 81, 232, 859, 780, 2131, 1, 15, 93, 499, 1091, 3821, 7953, 1, 16, 140, 592, 2774, 4912, 16556, 10864, 29681, 1, 19, 156, 1044, 3366

Offset: 1

Gary W. Adamson, Nov 18 2006

Row sums are powers of 4. The triangle is #4 in an infinite of generalized Pascal's triangles constrained by two rules: row sums are powers of N and upward sloping diagonals (as coefficients to polynomials with alternating signs) have roots N + 2*cos(2*Pi/Q).

Right border, A001835, and next to right border, A001353 = bisections of denominator of continued fraction [1, 2, 1, 2, 1, 2, 1, 2]; i.e., bisection of A002530. - Gary W. Adamson, Jun 21 2009

			First few rows of the triangle are:
  1;
  1,  3;
  1,  4, 11;
  1,  7, 15,  41;
  1,  8, 38,  56, 153;
  1, 11, 46, 186, 209, 571;
  1, 12, 81, 232, 859, 780, 2131;
  ...
The upward-sloping diagonal (1, 11, 38, 41) relates to the heptagon and in the form x^3 - 11x^2 + 38x - 41 has a root 5.24697960... = 4 + 2*cos(2*Pi/7). The corresponding matrix is [3, 1, 0; 1, 4, 1; 0, 1, 4]. The next upward-sloping diagonal relates to the octagon, with a characteristic polynomial x^3 - 12x^2 + 46x - 56 and a root 5.414213562... = 4 + 2*cos(2*Pi/8). The corresponding matrix is [4, 1, 0; 1, 4, 1; 0, 1, 4].

Cf. A125076, A125078.

Cf. A001835, A001353. - Gary W. Adamson, Jun 21 2009

Upward-sloping diagonals of the triangle are derived from (alternating) characteristic polynomials of two types of matrices: those of the form: (all 1's in the super and subdiagonals and 3,4,4,4,... in the main diagonal) and (all 1's in the super and subdiagonals and 4,4,4,... in the main diagonal.

A140827 Interleave denominators and numerators of convergents to sqrt(3).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 15, 26, 41, 56, 97, 153, 209, 362, 571, 780, 1351, 2131, 2911, 5042, 7953, 10864, 18817, 29681, 40545, 70226, 110771, 151316, 262087, 413403, 564719, 978122, 1542841, 2107560, 3650401, 5757961, 7865521, 13623482, 21489003, 29354524, 50843527, 80198051, 109552575

Offset: 0

Peter H van der Kamp, Jul 18 2008, Jul 22 2008

Coefficients of (1+r)^m modulo r^4-r^2+1.
The first few principal and intermediate convergents to 3^(1/2) are 1/1, 2/1, 3/2, 5/3, 7/4, 12/7; essentially, numerators=A143642 and denominators=A140827. - Clark Kimberling, Aug 27 2008
From Michel Dekking, Mar 11 2020: (Start)
This sequence can be seen as a generalization of the Fibonacci numbers A000045. The Zeckendorf expansion of a natural number uses the Fibonacci numbers as constituents. The Zeckendorf expansion is called a 2-bin decomposition in the paper by Demontigny et al.
The numbers a(n) are the constituents of the 3-bin decomposition of a natural number. See Example 4.2 and Proposition 4.3 in the Demontigny et al. paper.
Any natural number N can be uniquely expanded as
      N = Sum_{i=0..k} d(i)*a(i)
under the requirement d(i)d(i+1) = 0, and d(3i)d(3i+2) = 0 for all i.
Here k is the largest integer such that a(k) < N+1.
(End)

			(1+r)^(2+12*q)=(-1)^q*(a(1+18*q)*(1+r^2)+a(2+18*q)*r).
Here we write N = [d(k)d(k-1)...d(0)] for the 3-bin expansion of N.
0=[0], 1 =[1], 2=[10], 3=[100], 4=[1000], 5=[1001], 6=[1010], 7=[10000], 8=[10001], 9=[10010], 10=[10100], 11=[100000]. - _Michel Dekking_, Mar 11 2020

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Demontigny et al., Generalizing Zeckendorf's Theorem to f-decompositions, Journal of Number Theory 141, 135-158 (2014).
Peter H. van der Kamp, Global classification of two-component approximately integrable evolution equations, arXiv:0710.2233 [nlin.SI], 2007-2008.
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997), 122-126.
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-1).

Cf. A001075, A001835, A001353, A002965, A002530.

N:=100: a[0]:=1: a[1]:=1: for i from 2 to N do if i mod 3 = 1 then a[i]:=a[i-1]+a[i-3] else a[i]:=a[i-1]+a[i-2] fi od:

idnc[n_]:=Module[{cvrgts=Convergents[Sqrt[3],n],num,den},num=Take[ Numerator[ cvrgts],{2,-1,2}];den=Denominator[cvrgts]; Riffle[den, num,3]]; idnc[30] (* Harvey P. Dale, Mar 17 2012 *)

a(n) = 4*a(n-3) - a(n-6).
G.f.: ( 1+x+2*x^2-x^3-x^5 ) / ( 1-4*x^3+x^6 ).
a(n) = a(n-1)+a(n-3) if 3 |(n-1), else a(n)=a(n-1)+a(n-2), with n>1.
a(3*n-1) = A001075(n); a(3*n) = A001835(n-1); a(3*n+1) = A001353(n+1).
a(n)^2 = 1+3*a(n-1)^2 if n==2 (mod 3).

A192062 Square Array T(ij) read by antidiagonals (from NE to SW) with columns 2j being the denominators of continued fraction convergents to square root of (j^2 + 2j).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 4, 5, 3, 0, 1, 1, 5, 5, 11, 8, 1, 0, 1, 1, 6, 6, 19, 15, 13, 4, 0, 1, 1, 7, 7, 29, 24, 41, 21, 1, 0, 1, 1, 8, 8, 41, 35, 91, 56, 34, 5, 0, 1, 1, 9, 9, 55, 48, 169, 115, 153, 55, 1, 0, 1, 1, 10, 10, 71, 63, 281, 204, 436, 209, 89, 6

Offset: 0

Kenneth J Ramsey, Jun 21 2011

Column j=1 is the Fibonacci sequence A000045.  Column 2 is A002530; column 4 is A041011; column 6 is A041023; column 8 is A041039, column 10 is A041059, column 12 is A041083, column 14 is A041111 corresponding the denominators of continued fraction convergents to square root of 3,8,15,24,35,48 and 63.
  T(2*i-1,j)*T(2*i,j)^2*T(2*i+1,j)*j/2 appears to be always a triangular number, T(j*T(2*i,j)^2).
  T(2*i,j)*T(2*i+1,j)^2*T(2*i+2)*j/2 appears to always equal a triangular number, T(j*T(2*i,j)*T(2*i+2,j)).
Conjecture re relation of A192062 to the sequence of primes: T(2*n,j) = A(n,j)*T(n,j) where A(n,j) is from the square array A191971. There, A(3*n,j) = A(n,j)*B(n,j) where B(n,j) are integers. It appears further that B(5*n,j)=B(n,j)*C(n,j); C(7*n,j)= C(n,j)*D(n,j); D(11*n,j) = D(n,j)*E(n,j); E(13*n,j) = E(n,j)*F(n,j) and F(17*n,j) = F(n,j)*G(n,j) where C(n,j), D(n,j) etc. are all integers. My conjecture is that this property continues indefinitely and follows the sequence of primes.

			Array as meant by the definition
First column has index j=0
0  0  0   0   0   0   0 ...
1  1  1   1   1   1   1 ...
1  1  1   1   1   1   1 ...
1  2  3   4   5   6   7 ...
2  3  4   5   6   7   8 ...
1  5 11  19  29  41  55 ...
3  8 15  24  35  48  63 ...
1 13 41  91 169 281 433 ...
4 21 56 115 204 329 496 ...
.
.
.

Kenneth J Ramsey,Triangular and Fibonacci Numbers

Cf. A191579, A191971.

Each column j is a recursive sequence defined by T(0,j)=0, T(1,j) = 1, T(2i,j)= T(2i-2,j)+T(2i-1,j) and T(2i+1,j) = T(2i-1,j)+j*T(2i,j). Also, T(n+2,j) = (j+2)*T(n,j)-T(n-2,j).
T(2n,j) = Sum(k=1 to n) C(k)*T(2*k,j-1) where the C(k) are the n-th row of the triangle A191579.
T(2*i,j) = T(i,j)*A(i,j) where A(i,j) is from the table A(i,j) of A191971.
T(4*i,j) = (T(2*i+1)^2 - T(2*i-1)^2)/j
T(4*i+2,j) = T(2*i+2,j)^2 - T(2*i,j)^2

Corrected and edited by Olivier Gérard, Jul 05 2011

A041143 Denominators of continued fraction convergents to sqrt(80).

Original entry on oeis.org

1, 1, 17, 18, 305, 323, 5473, 5796, 98209, 104005, 1762289, 1866294, 31622993, 33489287, 567451585, 600940872, 10182505537, 10783446409, 182717648081, 193501094490, 3278735159921, 3472236254411, 58834515230497, 62306751484908, 1055742538989025

Offset: 0

N. J. A. Sloane

This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 16 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [First term 0 removed by _Georg Fischer_, Jul 01 2019]
Eric W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Cf. A041142, A020837, A040071, A010532, A001076, A002530.

List([0..30], n-> (5 +3*(-1)^n)*Fibonacci(3*(n+1))/16 ); # G. C. Greubel, Jul 02 2019

I:=[1,1,17,18]; [n le 4 select I[n] else 18*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

with(numtheory): cf := cfrac(sqrt(80),25): seq(nthdenom(cf,n), n=0..24); # Peter Luschny, Jul 06 2019

Denominator/@Convergents[Sqrt[80], 30] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 18 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,18,0]^n*[1;1;17;18])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

a(n) = (5 + 3*(-1)^n)*fibonacci(3*(n+1))/16 \\ Georg Fischer, Jul 01 2019

from sympy import continued_fraction_convergents as convergents, continued_fraction_iterator as cf, sqrt, denom
denominators = (denom(c) for c in convergents(cf(sqrt(80))))
print([next(denominators) for  in range(30)]) # _Ehren Metcalfe, Jul 03 2019

[(5 +3*(-1)^n)*fibonacci(3*(n+1))/16 for n in (0..30)] # G. C. Greubel, Jul 02 2019

G.f.: (1 + x - x^2) / (1 - 18*x^2 + x^4).
a(n) = 18*a(n-2) - a(n-4).
From Peter Bala, May 28 2014: (Start)
Let alpha = 2 + sqrt(5) and beta = 2 - sqrt(5) be the roots of the equation x^2 - 4*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n even, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n odd.
a(n) = A001076(n+1) for n even; a(n) = 1/4*A001076(n+1) for n odd.
a(n) = Product_{k = 1..floor(n/2)} ( 16 + 4*cos^2(k*Pi/(n+1)) ).
Recurrence equations: a(0) = 1, a(1) = 1 and for n >= 1, a(2*n) = 16*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = a(2*n) + a(2*n - 1). (End)
a(n) = (5 + 3*(-1)^n)*Fibonacci(3*(n+1))/16. - Ehren Metcalfe, Apr 15 2019

First term 0 removed from b-file, formulas and programs by Georg Fischer, Jul 01 2019

A041179 Denominators of continued fraction convergents to sqrt(99).

Original entry on oeis.org

1, 1, 19, 20, 379, 399, 7561, 7960, 150841, 158801, 3009259, 3168060, 60034339, 63202399, 1197677521, 1260879920, 23893516081, 25154396001, 476672644099, 501827040100, 9509559365899, 10011386405999, 189714514673881, 199725901079880, 3784780734111721

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 18 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

Cf. A041178, A010170, A020856, A010550, A002530.

I:=[1, 1, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Denominator[Convergents[Sqrt[99], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - Colin Barker, Nov 14 2013
a(n) = 20*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 12 2013
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 18 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 18*a(2*n) + a(2*n - 1). (End)

More terms from Colin Barker, Nov 14 2013

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