A134972 -id:A134972 - cp's OEIS frontend

A063727 a(n) = 2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136, 294912, 954368, 3088384, 9994240, 32342016, 104660992, 338690048, 1096024064, 3546808320, 11477712896, 37142659072, 120196169728, 388962975744, 1258710630400

Offset: 0

Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001

Essentially the same as A085449.
Convergents to 2*golden ratio = (1+sqrt(5)).
Number of ways to tile an n-board with two types of colored squares and four types of colored dominoes.
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 5 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(5). - Cino Hilliard, Sep 25 2005
a(n) is also the quasi-diagonal element A(i-1,i)=A(1,i-1) of matrix A(i,j) whose elements in first row A(1,k) and first column A(k,1) equal k-th Fibonacci Fib(k) and the generic element is the sum of adjacent (previous) in row and column minus the absolute value of their difference. - Carmine Suriano, May 13 2010
Equals INVERT transform of A006131: (1, 1, 5, 9, 29, 65, 181, ...). - Gary W. Adamson, Aug 12 2010
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 2's along the three central diagonals. - John M. Campbell, Jul 19 2011
The numbers composing the denominators of the fractional limit to A134972. - Seiichi Kirikami, Mar 06 2012
Pisano period lengths:  1, 1, 8, 1, 5, 8, 48, 1, 24, 5, 10, 8, 42, 48, 40, 1, 72, 24, 18, 5, ... - R. J. Mathar, Aug 10 2012

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 235.
John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Harry J. Smith)
J. Borowska and L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=2.
Index entries for linear recurrences with constant coefficients, signature (2,4).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006483, A103435, A006131, A269991.
Second row of A234357. Row sums of triangle A016095.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

List([0..25],n->2^n*Fibonacci(n+1)); # Muniru A Asiru, Nov 24 2018

[n le 2 select n else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 15 2008

a[n_]:=(MatrixPower[{{1,5},{1,1}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
CoefficientList[Series[1/(1 - 2 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 31 2014 *)
LinearRecurrence[{2, 4}, {1, 2}, 50] (* G. C. Greubel, Jan 07 2018 *)

s(n)=if(n<2,n+1,(s(n-1)+(s(n-2)*2))*2); for(n=0,32,print(s(n)))

{ for (n=0, 200, if (n>1, a=2*a1 + 4*a2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b063727.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 28 2009

[lucas_number1(n,2,-4) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = 2 * A087206(n+1).
From Vladeta Jovovic, Aug 16 2001: (Start)
a(n) = sqrt(5)/10*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)).
G.f.: 1/(1-2*x-4*x^2). (End)
From Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003: (Start)
a(2*n) = 4*a(n-1)^2 + a(n)^2.
A084057(n+1)/a(n) converges to sqrt(5). (End)
E.g.f.: exp(x)*(cosh(sqrt(5)*x)+sinh(sqrt(5)*x)/sqrt(5)). - Paul Barry, Sep 20 2003
a(n) = 2^n*Fibonacci(n+1). - Vladeta Jovovic, Oct 25 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*5^k. - Paul Barry, Nov 15 2003
a(n) = U(n, i/2)*(-i*2)^n, i^2=-1. - Paul Barry, Nov 17 2003
Simplified formula: ((1+sqrt(5))^n-(1-sqrt(5))^n)/sqrt(20). Offset 1. a(3)=8. - Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009
First binomial transform of 1,1,5,5,25,25. - Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009
a(n) = A(n-1,n) = A(n,n-1); A(i,j) = A(i-1,j) + A(i,j-1) - abs(A(i-1,j) - A(i,j-1)). - Carmine Suriano, May 13 2010
G.f.: G(0) where G(k) = 1 + 2*x*(1+2*x)/(1 - 2*x*(1+2*x)/(2*x*(1+2*x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
G.f.: G(0)/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 4*x )/( x*(4*k+4 + 4*x ) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013
Sum_{n>=0} 1/a(n) = A269991. - Amiram Eldar, Feb 01 2021

Better description from Jason Earls and Vladeta Jovovic, Aug 16 2001

Incorrect comment removed by Greg Dresden, Jun 02 2020

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Midsphere radius of regular icosahedron with unit edges.
Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014
Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014
This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - Wolfdieter Lang, Jan 08 2018
Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Feb 24 2021

			0.80901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014).
Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008).
Irit Dinur and David Steurer, Analytical approach to parallel repetition, arXiv:1305.1979 [cs.CC], 2013-2014.
Michael Penn, a golden value of cosine., YouTube video, 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

convert(sin(3*Pi/10),radical); # W. Edwin Clark, May 24 2023
Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[54 Degree],10,120][[1]] (* Harvey P. Dale, Apr 21 2018 *)

(1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - R. J. Mathar, Jun 18 2006
Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008
Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014
phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014
Equals 1/A134972. - R. J. Mathar, Jan 17 2021
Equals 2*A019836*A019872. - R. J. Mathar, Jan 17 2021
Equals (A094214 + 1)/2 or 1/(2*A094214). - Burak Muslu, Feb 24 2021
Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - Peter Bala, Mar 04 2022
Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - Amiram Eldar, Nov 28 2024
Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - Hugo Pfoertner, Nov 28 2024
Equals A134945/4. Root of 4*x^2-2*x-1=0. - R. J. Mathar, Aug 29 2025

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

A134945 Decimal expansion of 1 + sqrt(5).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

If "index" equals (0,2) then this sequence is the decimal expansion of (golden ratio divided by 5 = phi/5 = (1 + sqrt(5))/10). Example: 0.323606797...
Apart from the leading digit the same as A134972, A098317 and A002163. - R. J. Mathar, Aug 06 2013
Length of the longest diagonal in a regular 10-gon with unit side. - Mohammed Yaseen, Nov 12 2020
Abscissa of the first superstable point of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			3.2360679774997896964...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, p. 66.

Index entries for algebraic numbers, degree 2.

Cf. A000045, A002163, A098317, A134972.

RealDigits[1 + Sqrt[5], 10, 100][[1]] (* Michael De Vlieger, Nov 13 2020 *)

PARI
```
1 + sqrt(5) \\ Altug Alkan, Mar 19 2018
```

From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!+8*n!^2)/(n!^2*3^(2*n+2)).
Equals 1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
Equals 1/A019827. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} (1 + 1/Fibonacci(2*k)). - Amiram Eldar, May 27 2021

More terms from Jinyuan Wang, Mar 30 2020

A026273 a(n) = least k such that s(k) = n, where s = A026272.

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 57, 59, 61, 62, 64, 65, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 90, 91, 93, 95, 96, 98, 99

Offset: 1

Clark Kimberling

nonn

This is the lower s-Wythoff sequence, where s(n)=n+1.
See A184117 for the definition of lower and upper s-Wythoff sequences.  The first few terms of a and its complement, b=A026274, are obtained generated as follows:
s=(2,3,4,5,6,...);
a=(1,2,4,6,7,...)=A026273;
b=(3,5,8,11,13,...)=A026274.
Briefly:  b=s+a, and a=mex="least missing".
From Michel Dekking, Mar 12 2018: (Start)
One has r*(n-2*r+3) = n*r-2r^2+3*r = (n+1)*r-2.
So  a(n) = (n+1)*r-2, and we see that this sequence is simply the Beatty sequence of the golden ratio, shifted spatially and temporally. In other words: if w = A000201 = 1,3,4,6,8,9,11,12,14,...  is the lower Wythoff sequence, then a(n) = w(n+2) - 2.
(N.B. As so often, there is the 'offset 0 vs 1 argument', w = A000201 has offset 1; it would have been better to give (a(n)) offset 1, too).
This observation also gives an answer to Lenormand's question, and a simple proof of Mathar's conjecture in A059426.
(End)

Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 3.

Cf. A184117, A026274.

r=(1+Sqrt[5])/2;
a[n_]:=Floor[r*(n-2r+3)];
b[n_]:=Floor[r*r*(n+2r-3)];
Table[a[n],{n,200}]   (* A026273 *)
Table[b[n],{n,200}]   (* A026274 *)

a(n) = floor[r*(n-2*r+3)], where r=golden ratio.

b(n) = floor[(r^2)*(n+2*r-3)] = floor(n*A104457-A134972+1).

Extended by Clark Kimberling, Jan 14 2011

A096748 Expansion of (1+x)^2/(1-x^2-x^4).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 34, 42, 55, 68, 89, 110, 144, 178, 233, 288, 377, 466, 610, 754, 987, 1220, 1597, 1974, 2584, 3194, 4181, 5168, 6765, 8362, 10946, 13530, 17711, 21892, 28657, 35422, 46368, 57314, 75025, 92736, 121393, 150050

Offset: 0

Paul Barry, Jul 07 2004

The ratio a(n+1) / a(n) increasingly approximates two constants connected to the golden ratio phi = (1 + sqrt(5))/2: (phi+1)/2 = 1.30901699... = A239798 and (phi-1)*2 = 1.23606797... = A134972, according to whether n is odd or even. - Davide Rotondo, Jul 31 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Davide Rotondo, Perfino I Capelli Sono Tutti Contati (in Italian), see p. 11.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A079977.

Cf. A134972 and A239798 (limiting ratios for a(n+1)/a(n)).

CoefficientList[Series[(1+x)^2/(1-x^2-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,0,1},{1,2,2,2},50] (* Harvey P. Dale, Jan 29 2012 *)

a(n) = a(n-2) + a(n-4).
a(n) = 2*F((n+1)/2)*(1-(-1)^n)/2 + F((n+4)/2)*(1+(-1)^n)/2.
a(2*n) = A000045(n+2); a(2*n+1) = 2*A000045(n+1).
a(n) = Sum_{k=0..n} binomial(floor((n-k)/2), floor(k/2)). - Paul Barry, Jul 24 2004
a(n) = A079977(n) + A079977(n-2) + 2*A079977(n-1). - R. J. Mathar, Jul 15 2013

A134974 Decimal expansion of 4(-1 + phi) = 4A094214, where the golden ratio phi = A001622.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

This equals the dimensionless q-entropy (Tsallis entropy) of the set of 5 probabilities {p_i = 1/5, i = 1..5} for q = 1/2, which is S/k = -(1 - 5*(1/5)^(1/2))/(1 - 1/2) (k is the Boltzmann constant). See the Wikipedia link. - Wolfdieter Lang, Dec 06 2018

This constant - 2 = 2*sqrt(5) - 4 is the area of a regular pentagram formed by connecting the vertices of a unit-area regular pentagon. - Amiram Eldar, Nov 12 2021

			2.47213595499957939281834733746255247...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Wikipedia, Tsallis entropy.

Cf. A001622, A010476, A020762, A094214, A134972.

evalf[100](8/(1+sqrt(5))); # Muniru A Asiru, Dec 19 2018

RealDigits[4/GoldenRatio,10,120][[1]] (* Harvey P. Dale, Oct 30 2016 *)

2*(sqrt(5)-1) \\ or: digits( % \1e-35). - M. F. Hasler, Dec 14 2018

Equals 4*(-1 + phi) = 4*A094214, where phi = A001622. This is an integer in the field Q(sqrt(5)).
Equals 4/phi = 8/(1 + sqrt(5)).
Equals 10*A020762-2 = A010476-2. - R. J. Mathar, Oct 27 2008
Equals 2*(sqrt(5) - 1) = 2*A134972. - M. F. Hasler, Dec 14 2018

More terms from Harvey P. Dale, Oct 30 2016

Edited by Wolfdieter Lang, Dec 14 2018

A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.

Original entry on oeis.org

0, 4, 8, 32, 96, 320, 1024, 3328, 10752, 34816, 112640, 364544, 1179648, 3817472, 12353536, 39976960, 129368064, 418643968, 1354760192, 4384096256, 14187233280, 45910851584, 148570636288, 480784678912, 1555851902976, 5034842521600, 16293092655104

Offset: 0

Seiichi Kirikami, Mar 06 2012

a(n)/A063727(n) are convergents for A134972.

Abs(Sum_{i=0..n} C(n,n-i)*a(i)-(sqrt(5)-1)* A033887(n))->0. - Seiichi Kirikami, Jan 20 2016

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A063727, A033887, A085449, A103435, A134972, A269991.

Cf. A086344 (this sequence with signs).

I:=[0,4]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 16 2016

RecurrenceTable[{a[n]==2*a[n-1]+4*a[n-2], a[0]==0, a[1]==4}, a, {n, 30}]
LinearRecurrence[{2, 4}, {0, 4}, 40] (* Vincenzo Librandi, Jan 16 2016 *)

concat(0, Vec(4*x/(1-2*x-4*x^2) + O(x^40))) \\ Michel Marcus, Jan 16 2016

a(n) = (2/sqrt(5))*((1+sqrt(5))^n-(1-sqrt(5))^n).
G.f.: 4*x/(1-2*x-4*x^2). - Bruno Berselli, Mar 08 2012
a(n) = 4*A085449(n) = 2*A103435(n). - Bruno Berselli, Mar 08 2012
Sum_{n>=1} 1/a(n) = (1/4) * A269991. - Amiram Eldar, Feb 01 2021

A179999 Length of the n-th term in the modified Look and Say sequence A110393.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 10, 14, 18, 24, 30, 40, 50, 66, 82, 108, 134, 176, 218, 286, 354, 464, 574, 752, 930, 1218, 1506, 1972, 2438, 3192, 3946, 5166, 6386, 8360, 10334, 13528, 16722, 21890, 27058, 35420, 43782, 57312, 70842, 92734, 114626, 150048

Offset: 1

Nathaniel Johnston, Jan 13 2011

The average multiplicative growth from the n-th term to the (n+1)-st term is sqrt(phi) = 1.272..., where phi = (1+sqrt(5))/2 is the golden ratio, see A139339.

			The 6th term in A110393 is 21112211, so a(6) = 8.

Colin Barker, Table of n, a(n) for n = 1..1000
N. Johnston, Further Variants of the “Look-and-Say” Sequence
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Cf. A005341, A049194, A098596, A110393.

Cf. A001622, A134972, A139339, A239798.

CoefficientList[Series[((1+x) (-1-x+x^2) (1-x+x^2))/((1-x) (-1+x^2+x^4)),{x,0,99}],x] (* Peter J. C. Moses, Jun 23 2013 *)

Vec(x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)) + O(x^50)) \\ Colin Barker, Aug 10 2019

a(n) = length(A110393(n)).
From Colin Barker, Aug 10 2019: (Start)
G.f.: x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n>6. (End)
From A.H.M. Smeets, Aug 10 2019 (Start)
Limit_{n->oo} a(n+1)/a(n) = (1+phi)/2 = (3+sqrt(5))/4 = A239798 for odd n.
Limit_{n->oo} a(n+1)/a(n) = 2/phi = 4/(1+sqrt(5)) = A134972 for even n.
Limit_{n->oo} a(n+2)/a(n) = (1+phi)/phi = phi = A001622. (End)
For odd n > 1, a(n) = 4*Fibonacci((n + 1)/2) - 2. For even n, a(n) = 2*Fibonacci(n/2 + 2) - 2. - Ehren Metcalfe, Aug 10 2019

A229780 Decimal expansion of (3+sqrt(5))/10.

Original entry on oeis.org

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

Offset: 0

Joost Gielen, Sep 29 2013

sqrt((3+sqrt(5))/10) = sqrt(phi^2/5) = (5+sqrt(5))/10 = (3+sqrt(5))/10 + 2/10 = 0.723606797... .
Essentially the same as A134972, A134945, A098317 and A002163. - R. J. Mathar, Sep 30 2013
Equals one tenth of the limit of (G(n+2)+G(n+1)+G(n-1)+G(n-2))/G(n), where G(n) is any nonzero sequence satisfying the recurrence G(n+1) = G(n) + G(n-1) including A000032 and A000045, as n --> infinity. - Richard R. Forberg, Nov 17 2014
3+sqrt(5) is the perimeter of a golden rectangle with a unit width. - Amiram Eldar, May 18 2021
Constant x such that x = sqrt(x) - 1/5. - Andrea Pinos, Jan 15 2024

			0.5236067977499...

Cf. A094874, A187798.

RealDigits[GoldenRatio^2/5,10,120][[1]] (* Harvey P. Dale, Dec 02 2014 *)

(3+sqrt(5))/10 = (phi/sqrt(5))^2 = phi^2/5 where phi is the golden ratio.

cp's OEIS Frontend

A063727 a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

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A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

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A134945 Decimal expansion of 1 + sqrt(5).

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A026273 a(n) = least k such that s(k) = n, where s = A026272.

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A096748 Expansion of (1+x)^2/(1-x^2-x^4).

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A063727 a(n) = 2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

A134974 Decimal expansion of 4(-1 + phi) = 4A094214, where the golden ratio phi = A001622.

A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.