A318605 -id:A318605 - cp's OEIS frontend

A071101 Expansion of (5 + 6x + 3x^2 - 2x^3) / (1 - 2x - 2x^2 - 2x^3 + x^4) in powers of x.

5, 16, 45, 130, 377, 1088, 3145, 9090, 26269, 75920, 219413, 634114, 1832625, 5296384, 15306833, 44237570, 127848949, 369490320, 1067846845, 3086134658, 8919094697, 25776662080, 74495936025, 215297250946, 622220603405, 1798250918672, 5197041610021

Offset: 0

N. J. A. Sloane, May 28 2002

Number of tilings of the 2-mod-4 pillow of order n is a perfect square times a(n). [Propp, 1999, p. 272]

			G.f. = 5 + 16*x + 45*x^2 + 130*x^3 + 377*x^4 + 1088*x^5 + 3145*x^6 + 9090*x^7 + ...

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 12).

A.H.M. Smeets, Table of n, a(n) for n = 0..2169
J. Propp, Updated article
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A112835, A138573.

a:=[5,16,45,130];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2] +2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4) )); // G. C. Greubel, Jul 29 2019

seq(coeff(series((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

Table[Abs[Fibonacci[n+3, 1+I]]^2, {n,0,30}] (* Vladimir Reshetnikov, Oct 05 2016 *)
CoefficientList[Series[(5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 30}], x] (* Stefano Spezia, Sep 12 2018 *)
LinearRecurrence[{2,2,2,-1},{5,16,45,130},30] (* Harvey P. Dale, Oct 03 2024 *)

{a(n) = my(m = abs(n+3)); polcoeff( (x - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + x * O(x^m), m)};  /* Michael Somos, Dec 15 2011 */

x='x+O('x^33); Vec((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4)) \\ Altug Alkan, Sep 12 2018

from math import log
a0,a1,a2,a3,n = 130,45,16,5,3
print(0,a3)
print(1,a2)
print(2,a1)
print(3,a0)
while log(a0)/log(10) < 1000:
    a0,a1,a2,a3,n = 2*(a0+a1+a2)-a3,a0,a1,a2,n+1
    print(n,a0) # A.H.M. Smeets, Sep 12 2018

((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019

G.f.: (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).
a(-n) = a(-6 + n). a(-1) = 2, a(-2) = 1, a(-3) = 0. a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4). - Michael Somos, Dec 15 2011
A112835(2*n + 3) = a(n).
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

A138573 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

Original entry on oeis.org

0, 1, 2, 5, 16, 45, 130, 377, 1088, 3145, 9090, 26269, 75920, 219413, 634114, 1832625, 5296384, 15306833, 44237570, 127848949, 369490320, 1067846845, 3086134658, 8919094697, 25776662080, 74495936025, 215297250946, 622220603405

Offset: 0

Benoit Cloitre, May 12 2008

nonn

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 23 2008

Case P1 = 2, P2 = -4, Q = 1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 04 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kunle Adegoke, Robert Frontczak, and Taras Goy, Binomial Fibonacci sums from Chebyshev polynomials, J. Integer Seq., Vol. 26 (2023), Art. 23.9.6. See page 4; Preprint, arXiv:2308.04567 [math.CO], 2023.
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A071101, A000045, A100047, A138574, A143056, A272665.

a:=[0,1,2,5];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

seq(coeff(series((x*(1-x)*(x+1))/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

Round@Table[(((GoldenRatio + Sqrt[GoldenRatio])^n + (GoldenRatio - Sqrt[GoldenRatio])^n)/2 - (-1)^n Cos[n ArcTan[Sqrt[GoldenRatio]]])/Sqrt[5], {n, 0, 20}] (* or *) LinearRecurrence[{2, 2, 2, -1}, {0, 1, 2, 5}, 20] (* Vladimir Reshetnikov, May 11 2016 *)
Table[Abs[Fibonacci[n, 1 + I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 05 2016 *)
CoefficientList[Series[-x*(x-1)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

my(x='x+O('x^50)); concat([0], Vec(x*(1-x)*(1+x)/(1 -2*x -2*x^2 -2*x^3 +x^4))) \\ G. C. Greubel, Aug 08 2017

a(n) = round(w^n/2/sqrt(5)) where w = (1+r)/(1-r) = 2.89005363826396... and r = sqrt(sqrt(5)-2) = 0.485868271756...; for n >= 3, a(n) = A071101(n+3).
G.f.: -x*(x-1)*(1+x)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4). - R. J. Mathar, Jun 03 2009
From Peter Bala, Mar 04 2014: (Start)
Define a Lucas sequence {U(n)} in the ring of Gaussian integers by the recurrence U(n) = (1 + i)*U(n-1) + U(n-2) with U(0) = 0 and U(1) = 1. Then a(n) = |U(n)|^2.
Let a, b denote the zeros of x^2 - (1 + i)*x - 1 and c, d denote the zeros of x^2 - (1 - i)*x - 1.
Then a(n) = (a^n - b^n)*(c^n - d^n)/((a - b)*(c - d)).
a(n) = (alpha(1)^n + beta(1)^n - alpha(2)^n - beta(2)^n)/(2*sqrt(5)), where alpha(1), beta(1) are the roots of x^2 - ( 1 + sqrt(5))*x + 1 = 0, and alpha(2), beta(2) are the roots of x^2 - (1 - sqrt(5))*x + 1 = 0.
The o.g.f. is the Hadamard product of the rational functions x/(1 - (1 + i)x - x^2) and x/(1 - (1 - i)x - x^2). (End)
From Peter Bala, Mar 24 2014: (Start)
a(n) = (1/sqrt(5))*(T(n,phi) - T(n,-1/phi)), where phi = 1/2*(1 + sqrt(5)) is the golden ratio and T(n,x) denotes the Chebyshev polynomial of the first kind. Compare with the Fibonacci numbers, A000045, whose terms are given by the Binet formula 1/sqrt(5)*( phi^n - (-1/phi)^n ).
a(n) = top right (or bottom left) entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1]; the off-diagonal elements of M^n give the sequence of Fibonacci numbers. Bottom right entry of the matrix T(n, M) gives A138574. See the remarks in A100047 for the general connection between Chebyshev polynomials and linear divisibility sequences of the fourth order. (End)
a(n) = (((phi + sqrt(phi))^n + (phi - sqrt(phi))^n)/2 - (-1)^n * cos(n*arctan(sqrt(phi))))/sqrt(5), where phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, May 11 2016
a(n) = A143056(n+1)^2 + A272665(n+1)^2. - Vladimir Reshetnikov, Oct 05 2016
Limit_{n -> oo} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

A060790 Inscribe two circles of curvature 2 inside a circle of curvature -1. Sequence gives curvatures of the smallest circles that can be sequentially inscribed in such a diagram.

Original entry on oeis.org

-1, 2, 2, 3, 15, 38, 110, 323, 927, 2682, 7754, 22403, 64751, 187134, 540822, 1563011, 4517183, 13054898, 37729362, 109039875, 315131087, 910745750, 2632104062, 7606921923, 21984412383, 63536130986, 183622826522, 530679817859, 1533693138351, 4432455434478

Offset: 0

Brian Galebach, Apr 26 2001

The ratio of successive terms approaches the constant phi+sqrt(phi) ~= 2.89005363826396..., where phi is the golden ratio (sqrt(5)+1)/2. The ratio between the curvatures of two successively smaller circles approaches this constant in any apollonian packing as the curvatures increase.

For more comments, references and links, see A189226.

			After circles of 2, 2, 3, 15 have been inscribed in the diagram, the next smallest circle that can be inscribed has a curvature of 38.

Harry J. Smith, Table of n, a(n) for n = 0..200
Jeffrey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, arXiv:math/0101066 [math.MG], 2001.
Jeffrey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
I. Peterson, Circle Game, Science News, 4/21/01.
Index entries for linear recurrences with constant coefficients, signature (2, 2, 2, -1).

Cf. A042944, A189226, A189227.

a:=[-1,2,2,3];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

seq(coeff(series((x-1)*(1-3*x-3*x^2)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

CoefficientList[Series[(-3 z^3 + 4 z - 1)/(z^4 - 2 z^3 - 2 z^2 - 2 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{2, 2, 2, -1}, {-1, 2, 2, 3}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

{ for (n=0, 200, if (n>3, a=2*a1 + 2*a2 + 2*a3 - a4; a4=a3; a3=a2; a2=a1; a1=a, if (n==0, a=a4=-1, if (n==1, a=a3=2, if (n==2, a=a2=2, a=a1=3)))); write("b060790.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 12 2009

a(n) = 2a(n-1) + 2a(n-2) + 2a(n-3) - a(n-4).
G.f.: -(1-x)*(1 - 3*x - 3*x^2)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Apr 22 2012
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Corrected by T. D. Noe, Nov 08 2006

A071100 Expansion of (5 + 3x + x^2 - x^3) / (1 - 2x - 2x^2 - 2x^3 + x^4) in powers of x.

Original entry on oeis.org

5, 13, 37, 109, 313, 905, 2617, 7561, 21853, 63157, 182525, 527509, 1524529, 4405969, 12733489, 36800465, 106355317, 307372573, 888323221, 2567301757, 7419639785, 21443156953, 61971873769, 179102039257, 517614500173, 1495933669445, 4323328543981

Offset: 0

N. J. A. Sloane, May 28 2002

Number of tilings of the 0-mod-4 pillow of order n is a perfect square times a(n). [Propp, 1999, p. 271]

			G.f. = 5 + 13*x + 37*x^2 + 109*x^3 + 313*x^4 + 905*x^5 + 2617*x^6 + 7561*x^7 + ...

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 12).

A.H.M. Smeets, Table of n, a(n) for n = 0..2169
J. Propp, Updated article
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A112835.

a:=[5,13,37,109];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

seq(coeff(series((5+3*x+x^2-x^3)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

CoefficientList[Series[(5 + 3*x + x^2 -x^3)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4), {x, 0, 50}], x] (* Stefano Spezia, Sep 12 2018 *)
LinearRecurrence[{2,2,2,-1},{5,13,37,109},30] (* Harvey P. Dale, Sep 03 2021 *)

{a(n) = my(m = n+2); if( m < 0, m = -1 - m); polcoeff( (1 - x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + x * O(x^m), m)}; /* Michael Somos, Dec 15 2011 */

x='x+O('x^33); Vec((5+3*x+x^2-x^3)/(1-2*x-2*x^2-2*x^3+x^4)) \\ Altug Alkan, Sep 12 2018

G.f.: (5 + 3*x + x^2 -x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).
a(-n) = a(-5 + n). a(-1) = a(-2) = 1. a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4). - Michael Somos, Dec 15 2011
A112835(2*n + 2) = a(n).
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

A138574 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

Original entry on oeis.org

0, 1, 3, 9, 25, 73, 211, 609, 1761, 5089, 14707, 42505, 122841, 355017, 1026019, 2965249, 8569729, 24766977, 71577891, 206863945, 597847897, 1727812489, 4993470771, 14431398369, 41707515361, 120536956513, 348358269715, 1006774084809, 2909631106713, 8408989965961

Offset: 0

Benoit Cloitre, May 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A071101.

I:=[0,1,3,9,25]; [n le 5 select I[n] else 2*Self(n-1)+2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 14 2018

seq(coeff(series((x*(1+x+x^2-x^3))/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

LinearRecurrence[{2, 2, 2, -1}, {0, 1, 3, 9, 25}, 50] (* G. C. Greubel, Aug 08 2017 *)
CoefficientList[Series[x*( 1+x+x^2-x^3 )/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

x='x+O('x^50); concat([0], Vec(x*(1 +x +x^2 -x^3)/(1 -2*x -2*x^2 -2*x^3 +x^4))) \\ G. C. Greubel, Aug 08 2017

a(n) = round(w^n*(1 + 1/sqrt(5))/4) where w = (1+r)/(1-r) = 2.89005363826396... and r = sqrt(sqrt(5)-2) = 0.485868271756... .
G.f.: x*( 1 + x + x^2 - x^3 ) / ( 1 - 2*x - 2*x^2 - 2*x^3 + x^4 ). - R. J. Mathar, Jun 29 2013
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Terms corrected by Colin Barker, Jun 28 2013

A319129 Decimal expansion of (1 + sqrt(3) + sqrt(2*sqrt(3)))/2.

Original entry on oeis.org

2, 2, 9, 6, 6, 3, 0, 2, 6, 2, 8, 8, 6, 5, 3, 8, 2, 4, 5, 7, 0, 4, 9, 4, 1, 9, 1, 7, 7, 3, 6, 1, 7, 0, 2, 7, 1, 2, 2, 2, 6, 0, 6, 8, 5, 2, 5, 8, 2, 8, 4, 2, 6, 8, 9, 1, 2, 1, 8, 8, 0, 0, 0, 0, 8, 0, 4, 9, 2, 9, 9, 2, 2, 4, 5, 0, 3, 4, 8, 9, 8, 1

Offset: 0

A.H.M. Smeets, Sep 11 2018

This constant and its reciprocal are the real solutions of x^4 - 2*x^3 - 2*x + 1 = (x^2 - (sqrt(3)+1)*x + 1)*(x^2 + (sqrt(3)-1)*x + 1) = 0.

Decimal expansion of the largest x satisfying x^2 - (1 + sqrt(3))*x + 1 = 0.

			2.29663026288653824570494191773617027122260685258284268912188000080492992...

A.H.M. Smeets, Table of n, a(n) for n = 0..20000

Cf. A318605.

Digits:=100: evalf((1+sqrt(3)+sqrt(2*sqrt(3)))/2); # Muniru A Asiru, Sep 12 2018

RealDigits[(1 + Sqrt[3] + Sqrt[2 Sqrt[3]])/2, 10, 100][[1]] (* Bruno Berselli, Sep 13 2018 *)

(1+sqrt(3)+sqrt(2*sqrt(3)))/2 \\ Altug Alkan, Sep 11 2018

A038989 Expansion of (1 - x^2 - 2x^3) / (1 - 2x - 2x^2 - 2x^3 + x^4).

Original entry on oeis.org

1, 2, 5, 14, 41, 118, 341, 986, 2849, 8234, 23797, 68774, 198761, 574430, 1660133, 4797874, 13866113, 40073810, 115815461, 334712894, 967338217, 2795659334, 8079605429, 23350493066, 67484177441, 195032892538, 563655520661, 1628994688214, 4707882025385

Offset: 0

Colin Mallows

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

a:=[1,2,5,14];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

seq(coeff(series((1-x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

Vec((1 - x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + O(x^30)) \\ Colin Barker, Jul 16 2017

a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. - Colin Barker, Jul 16 2017

Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

A045821 Numerical distance between m-th and (n+m)-th circles in a loxodromic sequence of circles in which each 4 consecutive circles touch.

Original entry on oeis.org

-1, 1, 1, 1, 7, 17, 49, 145, 415, 1201, 3473, 10033, 28999, 83809, 242209, 700001, 2023039, 5846689, 16897249, 48833953, 141132743, 407881201, 1178798545, 3406791025, 9845808799, 28454915537, 82236232177, 237667122001

Offset: 0

Colin Mallows

Coxeter, H. S. M. "Numerical distances among the circles in a loxodromic sequence." Nieuw Archief voor Wiskunde 16 (1998): 1-10. (Note the word "circles" in the title!)

Muniru A Asiru, Table of n, a(n) for n = 0..500
H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.
H. S. M. Coxeter, Numerical distances among the spheres in a loxodromic sequence, Math. Intell. 19(4) 1997 pp. 41-47. (Note the word "spheres" in the title!) See page 45.
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A027674.

a:=[-1,1,1,1];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

with(combinat); F:=fibonacci;
f:=n->add(F(n-i)*binomial(n,2*(i-2)), i=2..n-1);
[seq(f(n),n=3..32)]; # Produces the sequence from a(3) onwards - N. J. A. Sloane, Sep 03 2018

CoefficientList[Series[-(x^3-x^2-3*x+1)/(x^4-2*x^3-2*x^2-2*x+1), {x, 0, 30}], x] (* Stefano Spezia, Sep 12 2018 *)

Vec(-(x^3-x^2-3*x+1)/(x^4-2*x^3-2*x^2-2*x+1) + O(x^100)) \\ Colin Barker, Sep 23 2013

a(n) = 2(a(n-1)+a(n-2)+a(n-3))-a(n-4).
a(n) = Sum{v=0 to [n/2]} binomial(n, 2v)*F(n-v-2) where F(m) is the m-th Fibonacci number.
G.f.: -(x^3-x^2-3*x+1) / (x^4-2*x^3-2*x^2-2*x+1). - Colin Barker, Sep 23 2013
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Reference and formulas from Floor van Lamoen

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A071101 Expansion of (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.

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A138573 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

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A060790 Inscribe two circles of curvature 2 inside a circle of curvature -1. Sequence gives curvatures of the smallest circles that can be sequentially inscribed in such a diagram.

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A071100 Expansion of (5 + 3*x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.

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A138574 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

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Extensions

A319129 Decimal expansion of (1 + sqrt(3) + sqrt(2*sqrt(3)))/2.

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A071101 Expansion of (5 + 6x + 3x^2 - 2x^3) / (1 - 2x - 2x^2 - 2x^3 + x^4) in powers of x.

A138573 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

A071100 Expansion of (5 + 3x + x^2 - x^3) / (1 - 2x - 2x^2 - 2x^3 + x^4) in powers of x.

A138574 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

A038989 Expansion of (1 - x^2 - 2x^3) / (1 - 2x - 2x^2 - 2x^3 + x^4).