A094214 -id:A094214 - cp's OEIS frontend

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

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

A167964 Signature sequence of phi^8 = 0.021286236252208..., where phi is A094214.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Index entries for sequences related to signature sequences

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*x, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N[# /. x -> 1/GoldenRatio^8] &] &)[[1 ;; terms]] /. x -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167965 Signature sequence of phi^7 = 0.034441853748633..., where phi is A094214.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Index entries for sequences related to signature sequences

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*x, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N[# /. x -> 1/GoldenRatio^7] &] &)[[1 ;; terms]] /. x -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167966 Signature sequence of phi^6 = 0.055728090000841..., where phi is the inverse golden ratio A094214.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167974.

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^6, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167967 Signature sequence of phi^5 = 0.090169943749474..., where phi is the inverse golden ratio A094214.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167973.

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^5, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Mar 08 2005

Only first term differs from the decimal expansion of phi.
Zelo extends work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace. - Jonathan Vos Post, Mar 02 2009 (Cf. last sentence in the Zelo reference. - Joerg Arndt, Jan 04 2014)
Hawkes asks: "What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?". - Charles R Greathouse IV, Dec 11 2012
This is the case n=10 in (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1+2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
An algebraic integer of degree 2, with minimal polynomial x^2 - 3x + 1. - Charles R Greathouse IV, Nov 12 2014 [The other root is 2 - phi = A132338 - Wolfdieter Lang, Aug 29 2022]
To eight digits: 5*(((Pi+1)/e)-1) = 2.61803395481182... - Dan Graham, Nov 21 2017
The ratio diagonal/side of the second smallest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020
phi^2/10 is the moment of inertia of a solid regular icosahedron with a unit mass and a unit edge length (see A341906). - Amiram Eldar, Jun 08 2021

			2.6180339887498948482045868343656381177203091798...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.17.1, p. 153.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Murray Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, Vol. 4, No. 2 (1966), pp. 157-162.
John Hawkes et al., Question 1029, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.
Casey Mongoven, Phi^2 number 1; electronic music created using Phi^2.
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
Damien Roy, Diophantine Approximation in Small Degree, arXiv:math/0303150 [math.NT], 2003.
Eric Weisstein's World of Mathematics, Chromatic Polynomial.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions.
Wikipedia, Perron number.
Dmitrij Zelo, Simultaneous Approximation to Real and p-adic Numbers, arXiv:0903.0086 [math.NT], 2009.
Index entries for algebraic numbers, degree 2.
Index entries for sequences related to moment of inertia.

Cf. A001622, A094214, A094885, A132338, A229780, A239798, A341906.
Cf. A001519, A001906, A032908, A049310.
2 + 2*cos(2*Pi/n): A116425 (n = 7), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

SetDefaultRealField(RealField(100)); (1+Sqrt(5))^2/4; // G. C. Greubel, Nov 23 2018

RealDigits[N[GoldenRatio+1,200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

(3+sqrt(5))/2 \\ Charles R Greathouse IV, Aug 21 2012

numerical_approx(golden_ratio^2, digits=100) # G. C. Greubel, Nov 23 2018

Equals 2 + A094214 = 1 + A001622. - R. J. Mathar, May 19 2008
Satisfies these three equations: x-sqrt(x)-1 = 0; x-1/sqrt(x)-2 = 0; x^2-3*x+1 = 0. - Richard R. Forberg, Oct 11 2014
Equals the nested radical sqrt(phi^2+sqrt(phi^4+sqrt(phi^8+...))). For a proof, see A094885. - Stanislav Sykora, May 24 2016
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (5*(2*n)!+8*n!^2)/(2*n!^2*3^(2*n+1)).
Equals 3/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Equals 1/A132338 = 2*A239798 = 5*A229780. - Mohammed Yaseen, Nov 04 2020
Equals Product_{k>=1} 1 + 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
c^n = phi * A001906(n) + A001519(n), where c = phi^2. - Gary W. Adamson, Sep 08 2023
Equals lim_{n->oo} S(n, 3)/S(n-1, 3) with the S-Chebyshev polynomials (see A049310), S(3, n) = A000045(2*(n+1)) = A001906(n+1). - Wolfdieter Lang, Nov 15 2023
From Peter Bala, May 08 2024: (Start)
Constant c = 2 + 2*cos(2*Pi/5).
The linear fractional transformation z -> c - c/z has order 5, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/z)))). (End)
Equals Product_{k>=1} (1 + 1/A032908(k)). - Amiram Eldar, Nov 28 2024

A022342 Integers with "even" Zeckendorf expansions (do not end with ...+F_2 = ...+1) (the Fibonacci-even numbers); also, apart from first term, a(n) = Fibonacci successor to n-1.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 105, 107

Offset: 1

Marc LeBrun

The Zeckendorf expansion of n is obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains; for example, 100 = 89 + 8 + 3.
The Fibonacci successor to n is found by replacing each F_i in the Zeckendorf expansion by F_{i+1}; for example, the successor to 100 is 144 + 13 + 5 = 162.
If k appears, k + (rank of k) does not (10 is the 7th term in the sequence but 10 + 7 = 17 is not a term of the sequence). - Benoit Cloitre, Jun 18 2002
From Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001: (Start)
a(n) = Sum_{k in A_n} F_{k+1}, where a(n)= Sum_{k in A_n} F_k is the (unique) expression of n as a sum of "noncontiguous" Fibonacci numbers (with index >= 2).
a(10^n) gives the first few digits of g = (sqrt(5)+1)/2.
The sequences given by b(n+1) = a(b(n)) obey the general recursion law of Fibonacci numbers. In particular the (sub)sequence (of a(-)) yielded by a starting value of 2=a(1), is the sequence of Fibonacci numbers >= 2. Starting points of all such subsequences are given by A035336.
a(n) = floor(phi*n+1/phi); phi = (sqrt(5)+1)/2. a(F_n)=F_{n+1} if F_n is the n-th Fibonacci number.
(End)
From Amiram Eldar, Sep 03 2022: (Start)
Numbers with an even number of trailing 1's in their dual Zeckendorf representation (A104326), i.e., numbers k such that A356749(k) is even.
The asymptotic density of this sequence is 1/phi (A094214). (End)

			The successors to 1, 2, 3, 4=3+1 are 2, 3, 5, 7=5+2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.
E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook)
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See pages 3, 6.
M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.
Jeffrey Shallit, The Hurt-Sada Array and Zeckendorf Representations, arXiv:2501.08823 [math.NT], 2025. See p. 2.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Jiemeng Zhang, Zhixiong Wen, and Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), Article P2.52.

Positions of 0's in A003849.
Complement of A003622.
Cf. A000201, A005206, A035336, A066096, A001950, A062879, A035516, A026274.
Cf. A035612, A094214, A104326, A356749.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

a022342 n = a022342_list !! (n-1)
a022342_list = filter ((notElem 1) . a035516_row) [0..]
-- Reinhard Zumkeller, Mar 10 2013

[Floor(n*(Sqrt(5)+1)/2)-1: n in [1..100]]; // Vincenzo Librandi, Feb 16 2015

A022342 := proc(n)
      local g;
      g := (1+sqrt(5))/2 ;
    floor(n*g)-1 ;
end proc: # R. J. Mathar, Aug 04 2013

With[{t=GoldenRatio^2},Table[Floor[n*t]-n-1,{n,70}]] (* Harvey P. Dale, Aug 08 2012 *)

PARI
```
a(n)=floor(n*(sqrt(5)+1)/2)-1
```

a(n)=(sqrtint(5*n^2)+n-2)\2 \\ Charles R Greathouse IV, Feb 27 2014

from math import isqrt
def A022342(n): return (n+isqrt(5*n**2)>>1)-1 # Chai Wah Wu, Aug 17 2022

a(n) = floor(n*phi^2) - n - 1 = floor(n*phi) - 1 = A000201(n) - 1, where phi is the golden ratio.
a(n) = A003622(n) - n. - Philippe Deléham, May 03 2004
a(n+1) = A022290(2*A003714(n)). - R. J. Mathar, Jan 31 2015
For n > 1: A035612(a(n)) > 1. - Reinhard Zumkeller, Feb 03 2015
a(n) = A000201(n) - 1. First differences are given in A014675 (or A001468, ignoring its first term). - M. F. Hasler, Oct 13 2017
a(n) = a(n-1) + 1 + A005614(n-2) for n > 1; also  a(n) = a(n-1) + A014675(n-2) = a(n-1) + A001468(n-1). - A.H.M. Smeets, Apr 26 2024

Name edited by Peter Munn, Dec 07 2021

A003401 Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285

Offset: 1

N. J. A. Sloane, R. K. Guy

The terms 1 and 2 correspond to degenerate polygons.
These are also the numbers for which phi(n) is a power of 2: A209229(A000010(a(n))) = 1. - Olivier Gérard Feb 15 1999
From Stanislav Sykora, May 02 2016: (Start)
The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 (A000079) are a subset and so are the Fermat primes (A019434), which are the only odd prime members.
The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).
Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)
If x,y are terms, and gcd(x,y) is a power of 2 then x*y is also a term. - David James Sycamore, Aug 24 2024

			34 is a term of this sequence because a circle can be divided into exactly 34 parts. 7 is not.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
Allan Clark, Elements of Abstract Algebra, Chapter 4, Galois Theory, Dover Publications, NY 1984, page 124.
Duane W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions." The American Mathematical Monthly 98.2 (1991): 97-108. - N. J. A. Sloane, Aug 05 2021
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.

Amiram Eldar, Table of n, a(n) for n = 1..10136 (terms below 10^100; terms 1..2000 from T. D. Noe)
Laura Anderson, Jasbir S. Chahal and Jaap Top, The last chapter of the Disquisitiones of Gauss, arXiv:2110.01355 [math.HO], 2021.
Wayne Bishop, How to construct a regular polygon, Amer. Math. Monthly 85(3) (1978), 186-188.
Alessandro Chiodo, A note on the construction of the Śrī Yantra, Sorbonne Université (Paris, France, 2020).
T. Chomette, Construction des polygones réguliers (in French).
Duane W. DeTemple, Carlyle circles and the Lemoine simplicity of polygon constructions, Amer. Math. Monthly 98(2) (1991), 97-108.
Bruce Director, Measurement and Divisibility.
David Eisenbud and Brady Haran, Heptadecagon and Fermat Primes (the math bit), Numberphile video (2015).
Mauro Fiorentini, Construibili (numeri).
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 463. Original (in Latin).
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag. 63(1) (1990), 3-20. [Annotated scanned copy] [DOI]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Johann Gustav Hermes, Über die Teilung des Kreises in 65537 gleiche Teile (About the division of the circle into 65537 equal pieces), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Vol. 3 (1894), 170-186.
Friedrich Julius Richelot, De resolutione algebraica aequationis X^257 = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata (On the resolution of the algebraic equation X^257 = 1, or ...), Journal für die reine und angewandte Mathematik 9 (1832), 1-26.
Pierre Wantzel, Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas (Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass), Journal de Mathématiques Pures et Appliquées 2 (1837), 366-372.
Eric Weisstein's World of Mathematics, Constructible Number.
Eric Weisstein's World of Mathematics, Constructible Polygon.
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Trigonometry.
Eric Weisstein's World of Mathematics, Trigonometry Angles.
Wikipedia, Constructible polygon.
Wikipedia, Johann Gustav Hermes.
Wikipedia, Friedrich Julius Richelot.
Wikipedia, Mohr-Mascheroni theorem.
Wikipedia, Pierre Wantzel.
Wikipedia, Poncelet-Steiner theorem.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.

Subsequence of A295298. - Antti Karttunen, Nov 27 2017
A004729 and A051916 are subsequences. - Reinhard Zumkeller, Mar 20 2010
Cf. A000079, A004169, A000215, A099884, A019434 (Fermat primes).
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17), A272536 (20).
Positions of zeros in A293516 (apart from two initial -1's), and in A336469, positions of ones in A295660 and in A336477 (characteristic function).
Cf. also A046528.
Cf. A000079, A092506.

a003401 n = a003401_list !! (n-1)
a003401_list = map (+ 1) $ elemIndices 1 $ map a209229 a000010_list
-- Reinhard Zumkeller, Jul 31 2012

Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ] (* Olivier Gérard Feb 15 1999 *)
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Take[ Union[ Flatten[ NestList[2# &, Times @@@ Table[ UnrankSubset[n, Join[{1}, Table[2^2^i + 1, {i, 0, 4}]]], {n, 63}], 11]]], 60] (* Robert G. Wilson v, Jun 11 2005 *)
nn=10; logs=Log[2,{2,3,5,17,257,65537}]; lim2=Floor[nn/logs[[1]]]; Sort[Reap[Do[z={i,j,k,l,m,n}.logs; If[z<=nn, Sow[2^z]], {i,0,lim2}, {j,0,1}, {k,0,1}, {l,0,1}, {m,0,1}, {n,0,1}]][[2,1]]]
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 1300; Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}] (* Robert G. Wilson v, Jul 28 2014 *)

for(n=1,10^4,my(t=eulerphi(n));if(t/2^valuation(t,2)==1,print1(n,", "))); \\ Joerg Arndt, Jul 29 2014

is(n)=n>>=valuation(n,2); if(n<7, return(n>0)); my(k=logint(logint(n,2),2)); if(k>32, my(p=2^2^k+1); if(n%p, return(0)); n/=p; unknown=1; if(n%p==0, return(0)); p=0; if(is(n)==0, 0, "unknown [has large Fermat number in factorization]"), 4294967295%n==0) \\ Charles R Greathouse IV, Jan 09 2022

is(n)=n>>=valuation(n,2); 4294967295%n==0 \\ valid for n <= 2^2^33, conjecturally valid for all n; Charles R Greathouse IV, Jan 09 2022

from sympy import totient
A003401_list = [n for n in range(1,10**4) if format(totient(n),'b').count('1') == 1]
# Chai Wah Wu, Jan 12 2015

Terms from 3 onward are computable as numbers such that cototient-of-totient equals the totient-of-totient: Flatten[Position[Table[co[eu[n]]-eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=m-eu[m]. - Labos Elemer, Oct 19 2001, clarified by Antti Karttunen, Nov 27 2017
Any product of 2^k and distinct Fermat primes (primes of the form 2^(2^m)+1). - Sergio Pimentel, Apr 30 2004, edited by Franklin T. Adams-Watters, Jun 16 2006
If the well-known conjecture that there are only five prime Fermat numbers F_k=2^{2^k}+1, k=0,1,2,3,4 is true, then we have exactly: Sum_{n>=1} 1/a(n)= 2*Product_{k=0..4} (1+1/F_k) = 4869735552/1431655765 = 3.40147098978.... - Vladimir Shevelev and T. D. Noe, Dec 01 2010
log a(n) >> sqrt(n); if there are finitely many Fermat primes, then log a(n) ~ k log n for some k. - Charles R Greathouse IV, Oct 23 2015

Definition clarified by Bill Gosper. - N. J. A. Sloane, Jun 14 2020

A047221 Numbers that are congruent to {2, 3} mod 5.

Original entry on oeis.org

2, 3, 7, 8, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 47, 48, 52, 53, 57, 58, 62, 63, 67, 68, 72, 73, 77, 78, 82, 83, 87, 88, 92, 93, 97, 98, 102, 103, 107, 108, 112, 113, 117, 118, 122, 123, 127, 128, 132, 133, 137, 138, 142, 143, 147, 148, 152, 153

Offset: 1

N. J. A. Sloane

Theorem: if 5^((n-1)/2) = -1 (mod n) then n == 2 or 3 (mod 5) (see Crandall and Pomerance).
Start with 2. The next number, 3, cannot be written as the sum of two of the previous terms. So 3 is in. 4=2+2, 5=2+3, 6=3+3, so these are not in. But you cannot obtain 7, so the next term is 7. And so on. - Fabian Rothelius, Mar 13 2001
Also numbers k such that k^2 == -1 (mod 5). - Vincenzo Librandi, Aug 05 2010
For any (t,s) < n, a(t)*a(s) != a(n) and a(t) - a(s) != a(n). - Anders Hellström, Jul 01 2015
These numbers appear in the product of a Rogers-Ramanujan identity. See A003106 also for references. - Wolfdieter Lang, Oct 29 2016

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 3.24, p. 154.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A118015 (floor(n^2/5)).
Cf. A003631 (primes), A094214.
Partitions into: A003106, A219607.

a047221 n = 5 * ((n - 1) `div` 2) + 3 - n `mod` 2
a047221_list = 2 : 3 : map (+ 5) a047221_list
-- Reinhard Zumkeller, Nov 27 2012

[ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];

{2,3}+#&/@(5 Range[0,30])//Flatten (* Harvey P. Dale, Jan 22 2023 *)

Vec(x*(2+x+2*x^2)/((1+x)*(1-x)^2) + O(x^80)) \\ Michel Marcus, Jun 30 2015

a(n) = 5*(n-1) - a(n-1) (with a(1)=2). - Vincenzo Librandi, Aug 05 2010
a(n) = (10*n - 3*(-1)^n - 5)/4.
G.f.: x*(2+x+2*x^2)/((1+x)*(1-x)^2).
a(n)^2 = 5*A118015(a(n)) + 4.
a(n) = 5 * (floor(n-1)/2) + 3 - n mod 2. - Reinhard Zumkeller, Nov 27 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/5. - Amiram Eldar, Dec 07 2021
E.g.f.: 2 + ((5*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 23 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/phi (A094214). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Apr 08 2002

Closed formula, g.f. and link added by Bruno Berselli, Nov 28 2010

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

cp's OEIS Frontend

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

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A167964 Signature sequence of phi^8 = 0.021286236252208..., where phi is A094214.

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A167965 Signature sequence of phi^7 = 0.034441853748633..., where phi is A094214.

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A167966 Signature sequence of phi^6 = 0.055728090000841..., where phi is the inverse golden ratio A094214.

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A167967 Signature sequence of phi^5 = 0.090169943749474..., where phi is the inverse golden ratio A094214.

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A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

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A022342 Integers with "even" Zeckendorf expansions (do not end with ...+F_2 = ...+1) (the Fibonacci-even numbers); also, apart from first term, a(n) = Fibonacci successor to n-1.

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