A243469 -id:A243469 - cp's OEIS frontend

A002530 a(n) = 4*a(n-2) - a(n-4) for n > 1, a(n) = n for n = 0, 1.

0, 1, 1, 3, 4, 11, 15, 41, 56, 153, 209, 571, 780, 2131, 2911, 7953, 10864, 29681, 40545, 110771, 151316, 413403, 564719, 1542841, 2107560, 5757961, 7865521, 21489003, 29354524, 80198051, 109552575, 299303201, 408855776, 1117014753, 1525870529, 4168755811

Offset: 0

N. J. A. Sloane

Denominators of continued fraction convergents to sqrt(3), for n >= 1.
Also denominators of continued fraction convergents to sqrt(3) - 1. See A048788 for numerators. - N. J. A. Sloane, Dec 17 2007. Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, ...
Consider the mapping f(a/b) = (a + 3*b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 5/3, 7/4, 19/11, ... converging to 3^(1/2). Sequence contains the denominators. The same mapping for N, i.e., f(a/b) = (a + Nb)/(a + b) gives fractions converging to N^(1/2). - Amarnath Murthy, Mar 22 2003
Sqrt(3) = 2/2 + 2/3 + 2/(3*11) + 2/(11*41) + 2/(41*153) + 2/(153*571), ...; the sum of the first 6 terms of this series = 1.7320490367..., while sqrt(3) = 1.7320508075... - Gary W. Adamson, Dec 15 2007
From Clark Kimberling, Aug 27 2008: (Start)
  Related convergents (numerator/denominator):
  lower principal convergents: A001834/A001835
  upper principal convergents: A001075/A001353
  intermediate convergents: A005320/A001075
  principal and intermediate convergents: A143642/A140827
  lower principal and intermediate convergents: A143643/A005246. (End)
Row sums of triangle A152063 = (1, 3, 4, 11, ...). - Gary W. Adamson, Nov 26 2008
From Alois P. Heinz, Apr 13 2011: (Start)
Also number of domino tilings of the 3 X (n-1) rectangle with upper left corner removed iff n is even.  For n=4 the 4 domino tilings of the 3 X 3 rectangle with upper left corner removed are:
   . ._. . ._. . ._. . ._.
   .|__| .|__| .| | | .|___|
   | |_| | | | | | ||| |_| |
   ||__| |||_| ||__| |_|_|  (End)
This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 2 and Q = -1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. - Peter Bala, Apr 18 2014
2^(-floor(n/2))*(1 + sqrt(3))^n = A002531(n) + a(n)*sqrt(3); integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 11 2018
Let T(n) = 2^(n mod 2), U(n) = a(n), V(n) = A002531(n), x(n) = V(n)/U(n). Then T(n*m) * U(n+m) = U(n)*V(m) + U(m)*V(n), T(n*m) * V(n+m) = 3*U(n)*U(m) + V(m)*V(n), x(n+m) = (3 + x(n)*x(m))/(x(n) + x(m)). - Michael Somos, Nov 29 2022

			Convergents to sqrt(3) are: 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530 for n >= 1.
1 + 1/(1 + 1/(2 + 1/(1 + 1/2))) = 19/11 so a(5) = 11.
G.f. = x + x^2 + 3*x^3 + 4*x^4 + 11*x^5 + 15*x^6 + 41*x^7 + ... - _Michael Somos_, Mar 18 2022

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
Russell Lyons, A bird's-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 181.
Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Harry J. Smith, Table of n, a(n) for n = 0..2000
Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Mario Catalani, Sequences related to convergents to square root of rationals, arXiv:math/0305270 [math.NT], 2003.
Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.
Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
MacTutor, D'Arcy Thompson on Greek irrationals
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
D'Arcy Thompson, Excess and Defect: Or the Little More and the Little Less, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 43-55 (13 pages). See page 48.
Hein van Winkel, Q-quadrangles inscribed in a circle, 2014. See Table 1. [Reference from Antreas Hatzipolakis, Jul 14 2014]
Russell Walsmith, CL-Chemy Transforms Fibonacci-Type Sequences to Arrays
E. W. Weisstein, MathWorld: Lehmer Number
Index entries for "core" sequences
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A002531 (numerators of convergents to sqrt(3)), A048788, A003297.
Bisections: A001353 and A001835.
Cf. A152063.
Cf. A108412, A049310.
Analog for sqrt(m): A000129 (m=2), A001076 (m=5), A041007 (m=6), A041009 (m=7), A041011 (m=8), A005668 (m=10), A041015 (m=11), A041017 (m=12), ..., A042935 (m=999), A042937 (m=1000).

I:=[0,1,1,3]; [n le 4 select I[n] else 4*Self(n-2) - Self(n-4): n in [1..50]]; // G. C. Greubel, Feb 25 2019

a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif n=3 then 3 else 4*a(n-2)-a(n-4) fi end; [ seq(a(i),i=0..50) ];
A002530:=-(-1-z+z**2)/(1-4*z**2+z**4); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Join[{0},Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3],n]]], {n,1,50}]] (* Stefan Steinerberger, Apr 01 2006 *)
Join[{0},Denominator[Convergents[Sqrt[3],50]]] (* or *) LinearRecurrence[ {0,4,0,-1},{0,1,1,3},50] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[n<0, -(-1)^n, 1] SeriesCoefficient[ x*(1+x-x^2)/(1-4*x^2+x^4), {x, 0, Abs@n}]; (* Michael Somos, Apr 18 2019 *)
a[ n_] := ChebyshevU[n-1, Sqrt[-1/2]]*Sqrt[2]^(Mod[n, 2]-1)/I^(n-1) //Simplify; (* Michael Somos, Nov 29 2022 *)

{a(n) = if( n<0, -(-1)^n * a(-n), contfracpnqn(vector(n, i, 1 + (i>1) * (i%2)))[2, 1])}; /* Michael Somos, Jun 05 2003 */

{ for (n=0, 50, a=contfracpnqn(vector(n, i, 1+(i>1)*(i%2)))[2, 1]; write("b002530.txt", n, " ", a); ); } \\ Harry J. Smith, Jun 01 2009

my(w=quadgen(12)); A002530(n)=real((2+w)^(n\/2)*if(bittest(n,0),1-w/3,w/3));
apply(A002530, [0..30]) \\ M. F. Hasler, Nov 04 2019

from functools import cache
@cache
def a(n): return [0, 1, 1, 3][n] if n < 4 else 4*a(n-2) - a(n-4)
print([a(n) for n in range(36)]) # Michael S. Branicky, Nov 13 2022

(x*(1+x-x^2)/(1-4*x^2+x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

G.f.: x*(1 + x - x^2)/(1 - 4*x^2 + x^4).
a(n) = 4*a(n-2) - a(n-4). [Corrected by László Szalay, Feb 21 2014]
a(n) = -(-1)^n * a(-n) for all n in Z, would satisfy the same recurrence relation. - Michael Somos, Jun 05 2003
a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1).
From Benoit Cloitre, Dec 15 2002: (Start)
a(2*n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/(2*sqrt(3)).
a(2*n) = A001353(n).
a(2*n-1) = ceiling((1 + 1/sqrt(3))/2*(2 + sqrt(3))^n) = ((3 + sqrt(3))^(2*n - 1) + (3 - sqrt(3))^(2*n - 1))/6^n.
a(2*n-1) = A001835(n). (End)
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n - k, k) * 2^floor((n - 2*k)/2). - Paul Barry, Jul 13 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2) + k, floor((n - 1)/2 - k))*2^k. - Paul Barry, Jun 22 2005
G.f.: (sqrt(6) + sqrt(3))/12*Q(0), where Q(k) = 1 - a/(1 + 1/(b^(2*k) - 1 - b^(2*k)/(c + 2*a*x/(2*x - g*m^(2*k)/(1 + a/(1 - 1/(b^(2*k + 1) + 1 - b^(2*k + 1)/(h - 2*a*x/(2*x + g*m^(2*k + 1)/Q(k + 1)))))))))). - Sergei N. Gladkovskii, Jun 21 2012
a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = 1/2*(sqrt(2) + sqrt(6)) and beta = (1/2)*(sqrt(2) - sqrt(6)). Cf. A108412. - Peter Bala, Apr 18 2014
a(n) = (-sqrt(2)*i)^n*S(n, sqrt(2)*i)*2^(-floor(n/2)) = A002605(n)*2^(-floor(n/2)), n >= 0, with i = sqrt(-1) and S the Chebyshev polynomials (A049310). - Wolfdieter Lang, Feb 10 2018
a(n+1)*a(n+2) - a(n+3)*a(n) = (-1)^n, n >= 0. - Kai Wang, Feb 06 2020
E.g.f.: sinh(sqrt(3/2)*x)*(sinh(x/sqrt(2)) + sqrt(2)*cosh(x/sqrt(2)))/sqrt(3). - Stefano Spezia, Feb 07 2020
a(n) = ((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*2^floor(n/2))/sqrt(3) = A002605(n)/2^floor(n/2). - Robert FERREOL, Apr 13 2023

Definition edited by M. F. Hasler, Nov 04 2019

A048788 a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2); a(n) = n for n = 0, 1.

Original entry on oeis.org

0, 1, 2, 3, 8, 11, 30, 41, 112, 153, 418, 571, 1560, 2131, 5822, 7953, 21728, 29681, 81090, 110771, 302632, 413403, 1129438, 1542841, 4215120, 5757961, 15731042, 21489003, 58709048, 80198051, 219105150, 299303201, 817711552, 1117014753

Offset: 0

Robin Trew (trew(AT)hcs.harvard.edu), Dec 11 1999

Numerators of continued fraction convergents to sqrt(3) - 1 (A160390). See A002530 for denominators. - N. J. A. Sloane, Dec 17 2007
Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, ... - Clark Kimberling, Sep 21 2013
A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. - Peter Bala, Jun 06 2014
From Sarah-Marie Belcastro, Feb 15 2022: (Start)
a(n) is also the number of perfect matchings of an edge-labeled 2 X (n-1) Mobius band grid graph, or equivalently the number of domino tilings of a 2 X (n-1) Mobius band grid. (The twist is on the length-n side.)
a(n) is also the output of Lu and Wu's formula for the number of perfect matchings of an m X n Mobius band grid, specialized to m = 2 with the twist on the length-n side.
2*a(n) is the number of perfect matchings of an edge-labeled 2 X (n-1) projective planar grid graph, or equivalently the number of domino tilings of a 2 X (n-1) projective planar grid. (End)

Russell Lyons, A bird's-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Sarah-Marie Belcastro, Domino Tilings of 2 X n Grids (or Perfect Matchings of Grid Graphs) on Surfaces, J. Integer Seq. 26 (2023), Article 23.5.6.
Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.
W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293 (2002), 235-246.
D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
J. L. Ramirez and F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=2, b=1.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002530, A002531.

Bisections are A001835 and A052530.

a:=[0,1,2,3];; for n in [5..40] do a[n]:=4a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019

I:=[0,1,2,3]; [n le 4 select I[n] else 4*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 10 2013

seq( simplify( `if`(`mod`(n,2)=0, 2*ChebyshevU((n-2)/2, 2), ChebyshevU((n-1)/2, 2) - ChebyshevU((n-3)/2, 2)) ), n=0..40); # G. C. Greubel, Dec 23 2019

Numerator[NestList[(2/(2 + #))&, 0, 40]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
CoefficientList[Series[x(1+2x-x^2)/(1-4x^2+x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
a0[n_]:= ((3+Sqrt[3])*(2-Sqrt[3])^n-((-3+Sqrt[3])*(2+Sqrt[3])^n))/6 // Simplify
a1[n_]:= 2*Sum[a0[i], {i, 1, n}]
Flatten[MapIndexed[{a1[#-1],a0[#]}&,Range[20]]] (* Gerry Martens, Jul 10 2015 *)
Round@Table[With[{r=1+Sqrt[2], s=1+Sqrt[3]}, ((r + (-1)^n/r) s^n/2^(n/2) - (1/r + (-1)^n r) 2^(n/2)/s^n) Sqrt[6]/12], {n, 0, 20}] (* or *) LinearRecurrence[ {0,4,0,-1}, {0,1,2,3}, 40] (* Vladimir Reshetnikov, May 11 2016 *)
Table[If[EvenQ[n], 2*ChebyshevU[(n-2)/2, 2], ChebyshevU[(n-1)/2, 2] - ChebyshevU[(n-3)/2, 2]], {n, 0, 40}] (* G. C. Greubel, Dec 23 2019 *)

main(size)=v=vector(size); v[1]=0;v[2]=1;v[3]=2;v[4]=3;for(i=5, size, v[i]=4*v[i-2] - v[i-4]); v; \\ Anders Hellström, Jul 11 2015

a=vector(50); a[1]=1; a[2]=2; for(n=3, #a, if(n%2==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]+a[n-2])); concat(0, a) \\ Colin Barker, Jan 30 2016

a(n)=([0,1,0,0;0,0,1,0;0,0,0,1;-1,0,4,0]^n*[0;1;2;3])[1,1] \\ Charles R Greathouse IV, Mar 16 2017

apply( {A048788(n)=imag((2+quadgen(12))^(n\/2)*if(bittest(n, 0), quadgen(12)-1, 2))}, [0..30]) \\ M. F. Hasler, Nov 04 2019

{a(n) = my(s=1,m=n); if(n<0,s=-(-1)^n; m=-n); polcoeff(x*(1+2*x-x^2)/(1-4*x^2+x^4) + x*O(x^m), m)*s}; /* Michael Somos, Sep 17 2020 */

@CachedFunction
def a(n):
    if (mod(n,2)==0): return 2*chebyshev_U((n-2)/2, 2)
    else: return chebyshev_U((n-1)/2, 2) - chebyshev_U((n-3)/2, 2)
[a(n) for n in (0..40)] # G. C. Greubel, Dec 23 2019

G.f.: x*(1+2*x-x^2)/(1-4*x^2+x^4). - Paul Barry, Sep 18 2009
a(n) = 4*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013
a(2*n-1) = A001835(n); a(2*n) = 2*A001353(n). - Peter Bala, Jun 06 2014
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a1(n-1),a0(n)] for n>0:
a0(n) = ((3+sqrt(3))*(2-sqrt(3))^n-((-3+sqrt(3))*(2+sqrt(3))^n))/6.
a1(n) = 2*Sum_{i=1..n} a0(i). (End)
a(n) = ((r + (-1)^n/r)*s^n/2^(n/2) - (1/r + (-1)^n*r)*2^(n/2)/s^n)*sqrt(6)/12, where r = 1 + sqrt(2), s = 1 + sqrt(3). - Vladimir Reshetnikov, May 11 2016
a(n) = 2*ChebyshevU(n-1, 2) if n is even and ChebyshevU(n, 2) - ChebyshevU(n-1, 2) if n in odd. - G. C. Greubel, Dec 23 2019
a(n) = -(-1)^n*a(-n) for all n in Z. - Michael Somos, Sep 17 2020

Denominator of g.f. corrected by Paul Barry, Sep 18 2009

Incorrect g.f. deleted by Colin Barker, Aug 10 2012

A108412 Expansion of (1 + x + x^2)/(1 - 4x^2 + x^4).

Original entry on oeis.org

1, 1, 5, 4, 19, 15, 71, 56, 265, 209, 989, 780, 3691, 2911, 13775, 10864, 51409, 40545, 191861, 151316, 716035, 564719, 2672279, 2107560, 9973081, 7865521, 37220045, 29354524, 138907099, 109552575, 518408351, 408855776, 1934726305

Offset: 0

Ralf Stephan, Jun 05 2005

This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 6 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence satisfies a linear recurrence of order four. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -6, 1, -6, ...] = 1/(1 - 1/(6 - 1/(1 - 1/(6 - ...)))) = 3 - sqrt(3) begins [0/1, 1/1, 6/5, 5/4, 24/19, 19/15, 90/71,...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [1, 6, 5, 24, 19, 90,...] is also a strong divisibility sequence. Cf. A005013 and A203976. - Peter Bala, May 19 2014
From Peter Bala, Mar 25 2018: (Start)
The following remarks assume an offset of 1.
Define a binary operation o on the real numbers by x o y = x*sqrt(1 + (1/2)*y^2) + y*sqrt(1 + (1/2)*x^2). The operation o is commutative and associative with identity 0. We have a(2*n + 1) = 1 o 1 o ... o 1 (2*n + 1 terms) and sqrt(6)*a(2*n) = (1 o 1 o ... o 1) (2*n terms). Cf. A005013 and A084068. For example, 1 o 1 = sqrt(6) and 1 o 1 o 1 = sqrt(6) o 1 = 5 = a(3).
From the obvious identity ( 1 o 1 o ... o 1 (2*n terms) ) o ( 1 o 1 o ... o 1 (2*m terms) ) = 1 o 1 o ... o 1 (2*n + 2*m terms) we find the relation a(2*n+2*m) = a(2*n)*sqrt(1 + 3*a(2*m)^2) + a(2*m)*sqrt(1 + 3*a(2*n)^2).
Similarly, from a(2*n+1) o a(2*m+1) = sqrt(6)*a(2*n+2*m+2) we find sqrt(6)*a(2*n+2*m+2) = a(2*n+1)*sqrt(1 + (1/2)*a(2*m+1)^2) + a(2*m+1)*sqrt(1 + (1/2)*a(2*n+1)^2). (End)

			G.f. = 1 + x + 5*x^2 + 4*x^3 + 19*x^4 + 15*x^5 + 71*x^6 + 56*x^7 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
P. Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
E. W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A001353, A001834.

Cf. A026741, A005013, A084068.

a := proc (n) if `mod`(n, 2) = 1 then 1/sqrt(2)*( ((sqrt(6) + sqrt(2))/2 )^n - ( (sqrt(6) - sqrt(2))/2 )^n) else 1/sqrt(12)*( ((sqrt(6) + sqrt(2))/2 )^n - ( (sqrt(6) - sqrt(2))/2 )^n) end if;
end proc:
seq(simplify(a(n)), n = 1..30); # Peter Bala, Mar 25 2018

CoefficientList[Series[(1+x+x^2)/(1-4x^2+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,4,0,-1},{1,1,5,4},40] (* Harvey P. Dale, Nov 15 2012 *)

{a(n) = my( w = quadgen(24)); simplify( polchebyshev( n, 2, w/2) / if( n%2, w, 1))}; /* Michael Somos, Feb 10 2015 */

a(0)=a(1)=1, a(2)=5, a(n)a(n+3) - a(n+1)a(n+2) = -1.
a(0)=1, a(1)=1, a(2)=5, a(3)=4, a(n) = 4*a(n-2)-a(n-4). - Harvey P. Dale, Nov 15 2012
a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = (1/2)*(sqrt(6) + sqrt(2)) (A188887) and beta = (1/2)*(sqrt(6) - sqrt(2)) (A101263). Equivalently, a(n) = U(n-1,sqrt(6)/2) for n odd and a(n) = (1/sqrt(6))*U(n-1,sqrt(6)/2) for n even, where U(n,x) is the Chebyshev polynomial of the second kind. - Peter Bala, Apr 18 2014
a(2*n) = A001834(n). a(2*n + 1) = A001353(n+1). - Michael Somos, Feb 10 2015
a(n) = -a(-2-n) for all n in Z. - Michael Somos, Feb 10 2015

A243470 Numerators of the rational convergents to the periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))).

Original entry on oeis.org

1, 7, 15, 112, 239, 1785, 3809, 28448, 60705, 453383, 967471, 7225680, 15418831, 115157497, 245733825, 1835294272, 3916322369, 29249550855, 62415424079, 466157519408, 994730462895, 7429270759673, 15853271982241, 118402174635360, 252657621252961, 1887005523406087

Offset: 1

Peter Bala, Jun 06 2014

The sequence of convergents to the simple periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))) begins [0/1, 1/2, 7/15, 15/32, 112/239, 239/510, ...]. The present sequence is the sequence of numerators of the convergents. It is a strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. The sequence is closely related to A041111, the Lehmer numbers U_n(sqrt(R),Q) with parameters R = 14 and Q = -1.

See A243469 for the sequence of denominators to the convergents.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Leonhard Euler, Introductio in analysin infinitorum, Vol.1, Chapter 18, section 378. French and German translations.
Eric Weisstein's World of Mathematics, Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Cf. A041111, A077412, A243469.

I:=[1,7,15,112]; [n le 4 select I[n] else 16*Self(n-2) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 21 2022

LinearRecurrence[{0,16,0,-1},{1,7,15,112},30] (* Harvey P. Dale, Nov 06 2017 *)

Vec(x*(1+7*x-x^2)/(1-16*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015

def b(n): return chebyshev_U(n,8) # b=A077412
def A243470(n): return 7*((n-1)%2)*b(n//2 -1) +(n%2)*(b((n-1)//2) -b((n-1)//2 -1))
[A243470(n) for n in (1..30)] # G. C. Greubel, May 21 2022

Let alpha = ( sqrt(14) + sqrt(18) )/2 and beta = ( sqrt(14) - sqrt(18) )/2 be the roots of the equation x^2 - sqrt(14)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = 7*(alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(2*n + 1) = Product_{k = 1..n} (14 + 4*cos^2(k*Pi/(2*n+1)));
a(2*n) = 7*Product_{k = 1..n-1} (14 + 4*cos^2(k*Pi/(2*n))).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 2, a(2*n) = 7*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 2*a(2*n) + a(2*n - 1).
Fourth-order recurrence: a(n) = 16*a(n - 2) - a(n - 4) for n >= 5.
O.g.f.: x*(1 + 7*x - x^2)/(1 - 16*x^2 + x^4).
a(2n-1) = A157456(n), a(2n) = 7*A077412(n-1). - Ralf Stephan, Jun 13 2014
a(n) = (1/2)*( 7*(1+(-1)^n)*ChebyshevU((n-2)/2, 8) + (1-(-1)^n)*(ChebyshevU((n- 1)/2, 8) - ChebyshevU((n-3)/2, 8)) ). - G. C. Greubel, May 21 2022

A145542 Numerators in continued fraction expansion of sqrt(3/5).

Original entry on oeis.org

1, 3, 7, 24, 55, 189, 433, 1488, 3409, 11715, 26839, 92232, 211303, 726141, 1663585, 5716896, 13097377, 45009027, 103115431, 354355320, 811826071, 2789833533, 6391493137, 21964312944, 50320119025, 172924670019, 396169459063, 1361433047208, 3119035553479

Offset: 1

Gary W. Adamson, Oct 12 2008

a(n)/A145543(n) tends to sqrt(3/5).

A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Related to the Lehmer sequence U_n(sqrt(R),Q) with parameters R = 6 and Q = -1. See A041023. - Peter Bala, Jun 06 2014

			[a(7), a(8)] = [433, 1488] X^4 * [1, 0] = [433, 1488].
a(5) = 55 = 2*a(4) + a(3) = 2*24 + 7.
G.f. = x + 3*x^2 + 7*x^3 + 24*x^4 + 55*x^5 + 189*x^6 + 433*x^7 + 1488*x^8 + ...

Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Eric Weisstein's World of Mathematics, Lehmer Number

Cf. A001090, A041203, A070997, A145543.

Numerator[Convergents[Sqrt[3/5], 30]] (* gives terms with 0 prepended *) (* Wesley Ivan Hurt, Jun 15 2014 *)

{a(n) = if( n<1, 0, polcoeff( x * (1 + 3*x - x^2) / (1 - 8*x^2 + x^4) + x * O(x^n), n))}; /* Michael Somos, Nov 14 2015 */

Numerators in continued fraction expansion of sqrt(3/5); i.e., of [1, 3, 2, 3, 2, 3, 2, 3, 2, ...].
[a(2*n - 1), a(2*n)] = X^n * [1,0], where X is the 2 X 2 matrix [1,2; 3,7].
Empirical G.f.: x*(1+3*x-x^2)/(1-8*x^2+x^4). - Colin Barker, Jan 04 2012
From Peter Bala, Jun 06 2014: (Start)
a(2*n + 1) = Product_{k=1..n} (6 + 4*cos^2(k*Pi/(2*n+1))).
a(2*n) = 3*Product_{k=1..n-1} (6 + 4*cos^2(k*Pi/(2*n))).
a(2*n + 1) = A070997(n); a(2*n) = 3*A001090(n). (End)

More terms from Wesley Ivan Hurt, Jun 15 2014

cp's OEIS Frontend

A002530 a(n) = 4*a(n-2) - a(n-4) for n > 1, a(n) = n for n = 0, 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Sage

Formula

Extensions

A048788 a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2); a(n) = n for n = 0, 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A108412 Expansion of (1 + x + x^2)/(1 - 4x^2 + x^4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A243470 Numerators of the rational convergents to the periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A145542 Numerators in continued fraction expansion of sqrt(3/5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula