A094214 -id:A094214 - cp's OEIS frontend

A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3.

1, 3, 16, 105, 715, 4896, 33553, 229971, 1576240, 10803705, 74049691, 507544128, 3478759201, 23843770275, 163427632720, 1120149658761, 7677619978603, 52623190191456, 360684711361585, 2472169789339635

Offset: 1

James R. Buddenhagen, Feb 06 2012

A Pell sequence related to Heronian triangles (rational triangles), see A206334. The connection is this: consider the problem of finding triangles with area a positive integer n, and with sides (a, b, n) where a, b are rational. Note that n is both the area and one side. For many values of n this is not possible, and the sequence of such numbers n is quite erratic (see A206334). Nonetheless, each term in this sequence is such a value of n. For example, for n = 105 you can take the other two sides, a and b, to be 10817/104, and 233/104 and the area will equal n, i.e., 105.

			G.f. = x + 3*x^2 + 16*x^3 + 105*x^4 + 715*x^5 + 4896*x^6 + 33553*x^7 + ... - _Michael Somos_, Jun 26 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Pridon Davlianidze, Problem B-1279, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 4 (2020), p. 368; An Unusual Generalization, Solution to Problem B-1279 by J. N. Senadheera, ibid., Vol. 59, No. 4 (2021), pp. 370-371.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A003482, A089508, A094214.

Subsequence of A206334.

a206351 n = a206351_list !! (n-1)
a206351_list = 1 : 3 : map (subtract 4)
               (zipWith (-) (map (* 7) (tail a206351_list)) a206351_list)
-- Reinhard Zumkeller, Feb 08 2012

[Fibonacci(2*n)*Fibonacci(2*n-3): n in [1..30]]; // G. C. Greubel, Aug 12 2018

genZ := proc(n)
local start;
option remember;
    start := [1, 3];
    if n < 3 then start[n]
    else 7*genZ(n - 1) - genZ(n - 2) - 4
    end if
end proc:
seq(genZ(n),n=1..20);

LinearRecurrence[{8, -8, 1}, {1, 3, 16}, 50] (* Charles R Greathouse IV, Feb 07 2012 *)
RecurrenceTable[{a[1] == 1, a[2] == 3, a[n] == 7 a[n - 1] - a[n - 2] - 4}, a, {n, 20}] (* Bruno Berselli, Feb 07 2012 *)
a[ n_] := Fibonacci[2 n] Fibonacci[2 n - 3]; (* Michael Somos, Jun 26 2018 *)
nxt[{a_,b_}]:={b,7b-a-4}; NestList[nxt,{1,3},20][[;;,1]] (* Harvey P. Dale, Aug 29 2024 *)

Vec((1-5*x)/(1-8*x+8*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 07 2012

{a(n) = fibonacci(2*n) * fibonacci(2*n - 3)}; /* Michael Somos, Jun 26 2018 */

From Bruno Berselli, Feb 07 2012: (Start)
G.f.: x*(1-5*x)/(1-8*x+8*x^2-x^3).
a(n) = A081018(n-1) + 1. (End)
a(n) = -A003482(-n) = Fibonacci(2*n)*Fibonacci(2*n-3). - Michael Somos, Jun 26 2018
a(n) = A089508(n-1) + 2 for n>1. - Bruno Berselli, Jun 20 2019 [Formula found by Umberto Cerruti]
Product_{n>=2} (1 - 1/a(n)) = 1/phi (A094214) (Davlianidze, 2020). - Amiram Eldar, Nov 30 2021
a(n) = (Fibonacci(2*n-2) + 1/Lucas(2*n-2))*(Fibonacci(2*n-1) + 1/Lucas(2*n-1)). - Peter Bala, Sep 03 2022

A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.
This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

			0.415823381635518674203484568810250332433169521255447672814363947...

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for sequences related to Chebyshev polynomials.
Index entries for algebraic numbers, degree 8.

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/15)
```

Equals 2*sin(Pi/m) for m=15, 2*A019821.
Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.
Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018

A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.99888450909488479883326824263012904463865119212705...

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176; Solution, ibid., Vol. 12, No. 1 (Winter 2000), pp. 61-62.
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139350.

RealDigits[Sin[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

sin((1+sqrt(5))/2) \\ Charles R Greathouse IV, May 13 2019

Equals sin(A001622).

Equals 1/A139350. - Amiram Eldar, Feb 07 2022

Leading zero removed by R. J. Mathar, Feb 05 2009

A035339 5th column of Wythoff array.

Original entry on oeis.org

8, 29, 42, 63, 84, 97, 118, 131, 152, 173, 186, 207, 228, 241, 262, 275, 296, 317, 330, 351, 364, 385, 406, 419, 440, 461, 474, 495, 508, 529, 550, 563, 584, 605, 618, 639, 652, 673, 694, 707, 728, 741, 762, 783, 796, 817, 838, 851, 872, 885, 906, 927, 940, 961

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), this sequence (k=5), A035340 (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(8*floor((n+1)*t)+5*n,n=0..80) ];

a[n_] := 8 * Floor[n * GoldenRatio] + 5*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A035340 6th column of Wythoff array.

Original entry on oeis.org

13, 47, 68, 102, 136, 157, 191, 212, 246, 280, 301, 335, 369, 390, 424, 445, 479, 513, 534, 568, 589, 623, 657, 678, 712, 746, 767, 801, 822, 856, 890, 911, 945, 979, 1000, 1034, 1055, 1089, 1123, 1144, 1178, 1199, 1233, 1267, 1288, 1322, 1356, 1377, 1411, 1432

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^7 = A094214^7 = 0.03444185... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), A035339 (k=5), this sequence (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(13*floor((n+1)*t)+8*n,n=0..80) ];

a[n_] := 13 * Floor[n * GoldenRatio] + 8*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n".

Original entry on oeis.org

8, 13, 21, 29, 34, 42, 47, 55, 63, 68, 76, 84, 89, 97, 102, 110, 118, 123, 131, 136, 144, 152, 157, 165, 173, 178, 186, 191, 199, 207, 212, 220, 228, 233, 241, 246, 254, 262, 267, 275, 280, 288, 296, 301, 309, 317, 322, 330, 335, 343, 351, 356, 364, 369, 377

Offset: 1

N. J. A. Sloane, Jan 26 2005

nonn

Let phi be the golden ratio. Using Fred Lunnon's formula in A101330 for Knuth's circle product, and the fact that phi^{-2} = 2-phi, plus [-x] = -[x]-1 for non-integer x, one obtains the formula below, expressing this sequence in terms of the lower Wythoff sequence. It follows in particular that the sequence of first differences 5,8,8,5,8,5,8,8,5,8,... of this sequence is the Fibonacci word A003849 on the alphabet {8,5}, shifted by 1. - Michel Dekking, Dec 23 2019
Also numbers with suffix string 0000, when written in Zeckendorf representation. - A.H.M. Smeets, Mar 20 2024
The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
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. 5.
W. F. Lunnon, Proof of formula, 2008.

Third row of array in A101330.
Cf. A101345 = Knuth's Fibonacci (or circle) product "2 o n".
Cf. A000201, A003849, A094214.

zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[ fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[i + j + 2], {i, Length[z1]}, {j, Length[z2]}]]; (* Robert G. Wilson v, Feb 04 2005 *)
Table[ kfp[3, n], {n, 50}] (* Robert G. Wilson v, Feb 04 2005 *)
Array[3*Floor[(# + 1)*GoldenRatio] + 2*# - 3 &, 100] (* Paolo Xausa, Mar 23 2024 *)

from math import isqrt
def A101642(n): return 3*(n+1+isqrt(5*(n+1)**2)>>1)+(n<<1)-3 # Chai Wah Wu, Aug 29 2022

From Michel Dekking, Dec 23 2019: (Start)
a(n) = 3*A000201(n+1) + 2n - 3.
a(n) = A101345(n) + A000201(n+1) + n + 1. (End)

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 04 2005

A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			-0.04722009625435983376687869404879456549554899472734...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139349.

Join[{0}, RealDigits[Cos[GoldenRatio], 10, 100][[1]]] (* Amiram Eldar, Feb 07 2022 *)

-cos((1+sqrt(5))/2) \\ Charles R Greathouse IV, May 13 2019

Equals 1/A139349. - Amiram Eldar, Feb 07 2022

Edited by N. J. A. Sloane, Dec 11 2008

A144749 Decimal expansion of the golden ratio powered to itself.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 20 2008

See A092134 for the continued fraction of this value, phi^phi, where phi = (sqrt(5)+1)/2 = A001622. - M. F. Hasler, Oct 08 2014

			Equals 2.178457567937599147372545702871245851807043301693254611347781924...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers.

Cf. A001622, A104457, A098317, A094214, A139339, A139340, A144713.

RealDigits[N[GoldenRatio^GoldenRatio,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

(t=(sqrt(5)+1)/2)^t \\ Use \p99 to get 99 digits; digits(%\.1^99) for the sequence of digits. - M. F. Hasler, Oct 08 2014

numerical_approx(golden_ratio^golden_ratio, digits=120) # G. C. Greubel, Jun 16 2022

Equals A001622^A001622.

A175288 Decimal expansion of the minimal positive constant x satisfying (cos(x))^2 = sin(x).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 23 2010, Mar 29 2010

This is the angle (in radians) at which the modified loop curve x^4=x^2*y-y^2 returns to the origin. Writing the curve in (r,phi) circular coordinates, r = sin(phi) * (cos^2(phi)-sin(phi)) /cos^4(phi), the two values of r=0 are phi=0 and the value of phi defined here. The equivalent angle of the Bow curve is Pi/4.

Also the minimum positive solution to tan(x) = cos(x). - Franklin T. Adams-Watters, Jun 17 2014

			x = 0.66623943.. = 38.1727076... degrees.

Eric Weisstein's World of Mathematics, Bow.

Cf. A019669, A094214, A001622, A195692, A352151.

r = 1/GoldenRatio;
N[ArcSin[r], 100]
RealDigits[%]  (* A175288 *)
RealDigits[x/.FindRoot[Cos[x]^2==Sin[x],{x,.6}, WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Nov 08 2011 *)
RealDigits[ ArcCos[ Sqrt[ (Sqrt[5] - 1)/2]], 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)

x = arcsin(A094214). cos(x)^2 = sin(x) = 0.618033988... = A094214.
From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A195692.
Equals arccos(1/sqrt(phi)).
Equals arctan(1/sqrt(phi)) = arccot(sqrt(phi)). (End)
Root of the equation cos(x) = tan(x). - Vaclav Kotesovec, Mar 06 2022

Disambiguated the curve here from the Mathworld bow curve - R. J. Mathar, Mar 29 2010

A298271 Expansion of x/((1 - x)(1 - 322x + x^2)).

Original entry on oeis.org

0, 1, 323, 104006, 33489610, 10783550415, 3472269744021, 1118060074024348, 360011871566096036, 115922704584208899245, 37326750864243699460855, 12019097855581887017496066, 3870112182746503375934272398, 1246164103746518505163818216091

Offset: 0

Bruno Berselli, Jan 16 2018

Colin Barker, Table of n, a(n) for n = 0..399
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A000045, A253368, A298101.

CoefficientList[x/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]

makelist(coeff(taylor(x/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);

a(n)=([0,1,0; 0,0,1; 1,-323,323]^n*[0;1;323])[1,1] \\ Charles R Greathouse IV, Jan 18 2018

concat(0, Vec(x / ((1 - x)*(1 - 322*x + x^2)) + O(x^15))) \\ Colin Barker, Jan 19 2018

gf = x/((1-x)*(1-322*x+x^2))
print(taylor(gf, x, 0, 20).list())

G.f.: x/((1 - x)*(1 - 322*x + x^2)).
a(n) = a(-n-1) = 323*a(n-1) - 323*a(n-2) + a(n-3).
a(n) = (1/5760)*((2 + sqrt(5))^(4*n+2) + (2 + sqrt(5))^-(4*n+2) - 18).
a(n) = A298101(n) - A298101(n-1) + A298101(n-2) - A298101(n-3) + ..., hence:
a(n) + a(n-1) = A298101(n).
a(n) - a(n-1) = (1/144)*Fibonacci(12*n).
a(n) - a(n-2) = (1/8)*Fibonacci(12*n-6).
a(n)*a(n-2) = a(n-1)*(a(n-1) - 1).
Sum_{j>1} 1/a(j) = 161 - 72*sqrt(5) = A094214^12.
a(n) = A157459(n+1)/72. - Greg Dresden, Dec 02 2021

cp's OEIS Frontend

A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3.

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A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

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A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

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A035339 5th column of Wythoff array.

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A035340 6th column of Wythoff array.

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A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n".

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A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

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A298271 Expansion of x/((1 - x)(1 - 322x + x^2)).