A231187 -id:A231187 - cp's OEIS frontend

A262773 A Beatty sequence: a(n)=floor(q*n) where q=A231187.

0, 2, 4, 6, 8, 11, 13, 15, 17, 20, 22, 24, 26, 29, 31, 33, 35, 38, 40, 42, 44, 47, 49, 51, 53, 56, 58, 60, 62, 65, 67, 69, 71, 74, 76, 78, 80, 83, 85, 87, 89, 92, 94, 96, 98, 101, 103, 105, 107, 110, 112, 114, 116, 119, 121, 123, 125, 128, 130, 132, 134, 137, 139

Offset: 0

Patrick D McLean, Sep 30 2015

nonn

Beatty sequence of the longer diagonal (A231187) in a regular heptagon with sidelength 1.

Complement of Beatty sequence A262770 of the longer diagonal (A160389) in a regular heptagon with sidelength 1.

Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997
Index entries for sequences related to Beatty sequences

Complement of A262770.

Table[Floor[n/(2 Cos[3 Pi/7])], {n, 0, 106}] (* Michael De Vlieger, Oct 05 2015 *)

q=roots([1,-2,-1,1])(1); a(n)=floor(q*n)

a(n) = floor(n/(2*cos(3*Pi/7))) \\ Michel Marcus, Oct 05 2015

A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 15, 35, 28, 9, 1, 1, 21, 70, 84, 45, 11, 1, 1, 28, 126, 210, 165, 66, 13, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1

Offset: 0

Philippe Deléham, Aug 14 2003

Coefficient array for Morgan-Voyce polynomial b(n,x). A053122 (unsigned) is the coefficient array for B(n,x). Reversal of A054142. - Paul Barry, Jan 19 2004
This triangle is formed from even-numbered rows of triangle A011973 read in reverse order. - Philippe Deléham, Feb 16 2004
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k+1 peaks. T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k peaks at height >= 2. T(n,k) is the number of directed column-convex polyominoes of area n+1, having k+1 columns. - Emeric Deutsch, May 31 2004
Riordan array (1/(1-x), x/(1-x)^2). - Paul Barry, May 09 2005
The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse of A039599. - Philippe Deléham, May 26 2005
The n-th row gives absolute values of coefficients of reciprocal of g.f. of bottom-line of n-wave sequence. - Floor van Lamoen (fvlamoen(AT)planet.nl), Sep 24 2006
Unsigned version of A129818. - Philippe Deléham, Oct 25 2007
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k >=1 (height(alpha) = |Im(alpha)|) and of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Oct 02 2008
A085478 is jointly generated with A078812 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n>1, u(n,x) = u(n-1,x)+x*v(n-1)x and v(n,x) = u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012
Per Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016
Subtriangle of the triangle given by (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
T(n,k) is also the number of compositions (ordered partitions) of 2*n+1 into 2*k+1 parts which are all odd. Proof: The o.g.f. of column k, x^k/(1-x)^(2*k+1) for k >= 0, is the o.g.f. of the odd-indexed members of the sequence with o.g.f. (x/(1-x^2))^(2*k+1) (bisection, odd part). Thus T(n,k) is obtained from the sum of the multinomial numbers A048996 for the partitions of 2*n+1 into 2*k+1 parts, all of which are odd. E.g., T(3,1) = 3 + 3 from the numbers for the partitions [1,1,5] and [1,3,3], namely 3!/(2!*1!) and 3!/(1!*2!), respectively. The number triangle with the number of these partitions as entries is A152157. - Wolfdieter Lang, Jul 09 2012
The matrix elements of the inverse are T^(-1)(n,k) = (-1)^(n+k)*A039599(n,k). - R. J. Mathar, Mar 12 2013
T(n,k) = A258993(n+1,k) for k = 0..n-1. - Reinhard Zumkeller, Jun 22 2015
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the algebraic function F(x)*G(x)^n about 0, where F(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) and G(x) = ((1 + sqrt(1 + 4*x))/2)^2. For example, for n = 4, (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) * ((1 + sqrt(1 + 4*x))/2)^8 = (x^4  + 10*x^3 + 15*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 23 2018
Row n also gives the coefficients of the characteristc polynomial of the tridiagonal n X n matrix M_n given in A332602: Phi(n, x) := Det(M_n - x*1_n) = Sum_{k=0..n} T(n, k)*(-x)^k, for n >= 0, with Phi(0, x) := 1. - Wolfdieter Lang, Mar 25 2020
It appears that the largest root of the n-th degree polynomial is equal to the sum of the distinct diagonals of a (2*n+1)-gon including the edge, 1. The largest root of x^3 - 6*x^2 + 5*x - 1 is 5.048917... = the sum of (1 + 1.80193... + 2.24697...). Alternatively, the largest root of the n-th degree polynomial is equal to the square of sigma(2*n+1). Check: 5.048917... is the square of sigma(7), 2.24697.... Given N = 2*n+1, sigma(N) (N odd) can be defined as 1/(2*sin(Pi/(2*N))). Relating to the 9-gon, the largest root of x^4 - 10*x^3 + 15*x^2 - 7*x + 1 is 8.290859..., = the sum of (1 + 1.879385... + 2.532088... + 2.879385...), and is the square of sigma(9), 2.879385... Refer to A231187 for a further clarification of sigma(7). - Gary W. Adamson, Jun 28 2022
For n >=1, the n-th row is given by the coefficients of the minimal polynomial of -4*sin(Pi/(4*n + 2))^2. - Eric W. Weisstein, Jul 12 2023
Denoting this lower triangular array by L, then L * diag(binomial(2*k,k)^2) * transpose(L) is the LDU factorization of A143007, the square array of crystal ball sequences for the A_n X A_n lattices. - Peter Bala, Feb 06 2024
T(n, k) is the number of occurrences of the periodic substring (01)^k in the periodic string (01)^n (see Proposition 4.7 at page 7 in Fang). - Stefano Spezia, Jun 09 2024

			Triangle begins as:
  1;
  1    1;
  1    3    1;
  1    6    5    1;
  1   10   15    7    1;
  1   15   35   28    9    1;
  1   21   70   84   45   11    1;
  1   28  126  210  165   66   13    1;
  1   36  210  462  495  286   91   15    1;
  1   45  330  924 1287 1001  455  120   17    1;
  1   55  495 1716 3003 3003 1820  680  153   19    1;
...
From _Philippe Deléham_, Mar 26 2012: (Start)
(0, 1, 0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 1,  1
  0, 1,  3,   1
  0, 1,  6,   5,   1
  0, 1, 10,  15,   7,   1
  0, 1, 15,  35,  28,   9,  1
  0, 1, 21,  70,  84,  45, 11,  1
  0, 1, 28, 126, 210, 165, 66, 13, 1. (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - _N. J. A. Sloane_, May 10 2012
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
Paul Barry and Aoife Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
Eduardo H. M. Brietzke, Recurrence Relations for Sums of Binomial Coefficients and Some Generalizations, Journal of Integer Sequences, Vol. 27 (2024), Article 24.3.4.
E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807, Theorem 3.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62. - _Abdullahi Umar_, Oct 02 2008
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Row sums: A001519. Signed versions: A123970, A129818.
Cf. A000108, A001006, A007318, A098158, A183160, A078812, A091042.
Cf. A258993, A025581, A051162, A054142, A332602.
Cf. A231187, A143007.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k, 2*k) ))); # G. C. Greubel, Aug 01 2019

a085478 n k = a085478_tabl !! n !! k
a085478_row n = a085478_tabl !! n
a085478_tabl = zipWith (zipWith a007318) a051162_tabl a025581_tabl
-- Reinhard Zumkeller, Jun 22 2015

[Binomial(n+k, 2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T := (n,k) -> binomial(n+k,2*k): seq(seq(T(n,k), k=0..n), n=0..11);

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A078812 *) (*Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n + k, 2 k], {n, 0, 12}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 01 2019 *)
CoefficientList[Table[Fibonacci[2 n + 1, Sqrt[x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Jul 03 2023 *)
Join[{{1}}, CoefficientList[Table[MinimalPolynomial[-4 Sin[Pi/(4 n + 2)]^2, x], {n, 20}], x]] (* Eric W. Weisstein, Jul 12 2023 *)

PARI
```
T(n,k) = binomial(n+k,n-k)
```

[[binomial(n+k,2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n, k) = (n+k)!/((n-k)!*(2*k)!).
G.f.: (1-z)/((1-z)^2-tz). - Emeric Deutsch, May 31 2004
Row sums are A001519 (Fibonacci(2n+1)). Diagonal sums are A011782. Binomial transform of A026729 (product of lower triangular matrices). - Paul Barry, Jun 21 2004
T(n, 0) = 1, T(n, k) = 0 if n=0} T(n-1-j, k-1)*(j+1). T(0, 0) = 1, T(0, k) = 0 if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j). For the column k, g.f.: Sum_{n>=0} T(n, k)*x^n = (x^k) / (1-x)^(2*k+1). - Philippe Deléham, Feb 15 2004
Sum_{k=0..n} T(n,k)*x^(2*k) = A000012(n), A001519(n+1), A001653(n), A078922(n+1), A007805(n), A097835(n), A097315(n), A097838(n), A078988(n), A097841(n), A097727(n), A097843(n), A097730(n), A098244(n), A097733(n), A098247(n), A097736(n), A098250(n), A097739(n), A098253(n), A097742(n), A098256(n), A097767(n), A098259(n), A097770(n), A098262(n), A097773(n), A098292(n), A097776(n) for x=0,1,2,...,27,28 respectively. - Philippe Deléham, Dec 31 2007
T(2*n,n) = A005809(n). - Philippe Deléham, Sep 17 2009
A183160(n) = Sum_{k=0..n} T(n,k)*T(n,n-k). - Paul D. Hanna, Dec 27 2010
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 06 2012
O.g.f. for column k: x^k/(1-x)^(2*k+1), k >= 0. [See the o.g.f. of the triangle above, and a comment on compositions. - Wolfdieter Lang, Jul 09 2012]
E.g.f.: (2/sqrt(x + 4))*sinh((1/2)*t*sqrt(x + 4))*cosh((1/2)*t*sqrt(x)) = t + (1 + x)*t^3/3! + (1 + 3*x + x^2)*t^5/5! + (1 + 6*x + 5*x^2 + x^3)*t^7/7! + .... Cf. A091042. - Peter Bala, Jul 29 2013
T(n, k) = A065941(n+3*k, 4*k) = A108299(n+3*k, 4*k) = A194005(n+3*k, 4*k). - Johannes W. Meijer, Sep 05 2013
Sum_{k=0..n} (-1)^k*T(n,k)*A000108(k) = A000007(n) for n >= 0. - Werner Schulte, Jul 12 2017
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A001006(n) for n >= 0. - Werner Schulte, Jul 12 2017
From  Peter Bala, Jun 26 2025: (Start)
The n-th row polynomial b(n, x) = (-1)^n * U(2*n, (i/2)*sqrt(x)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.
b(n, x) = (-1)^n * Dir(n, -1 - x/2), where Dir(n, x) is the n-th row polynomial of the triangle A244419.
b(n, -1 - x) is the n-th row polynomial of A098493. (End)

A160389 Decimal expansion of 2*cos(Pi/7).

Original entry on oeis.org

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

Offset: 1

Harry J. Smith, May 31 2009

Arises in the approximation of 14-fold quasipatterns by 14 Fourier modes.
Let DTS(n^c) denote the set of languages accepted by a deterministic Turing machine with space n^(o(1)) and time n^(c+o(1)), and let SAT denote the Boolean satisfiability problem. Then (1) SAT is not in DTS(n^c) for any c < 2*cos(Pi/7), and (2) the Williams inference rules cannot prove that SAT is not in DTS(n^c) for any c >= 2*cos(Pi/7). These results also apply to the Boolean satisfiability problem mod m where m is in A085971 except possibly for one prime. - Charles R Greathouse IV, Jul 19 2012
rho(7):= 2*cos(Pi/7) is the length ratio (smallest diagonal)/side in the regular 7-gon (heptagon). The algebraic number field Q(rho(7)) of degree 3 is fundamental for the 7-gon. See A187360 for the minimal polynomial C(7, x) of rho(7). The other (larger) diagonal/side ratio in the heptagon is sigma(7) = -1 + rho(7)^2, approx. 2.2469796. (see the decimal expansion in A231187). sigma(7) is the limit of a(n+1)/a(n) for n->infinity for the sequences like A006054 and A077998 which can be considered as analogs of the Fibonacci sequence in the pentagon. Thus sigma(7) plays in the heptagon the role of the golden section in the pentagon. See the P. Steinbach reference. - Wolfdieter Lang, Nov 21 2013
An algebraic integer of degree 3 with minimal polynomial x^3 - x^2 - 2x + 1. - Charles R Greathouse IV, Nov 12 2014
The other two solutions of the minimal polynomial of rho(7) = 2*cos(Pi/7) are 2*cos(3*Pi/7) and 2*cos(5*Pi/7). See eq. (20) of the W. Lang link. - Wolfdieter Lang, Feb 11 2015
The constant is the square root of 3.24697... (cf. A116425). It is the fifth-longest diagonal in the regular 14-gon with unit radius, which equals 2*sin(5*Pi/14). - Gary W. Adamson, Feb 14 2022

			1.801937735804838252472204639014890102331838324263714300107124846398864...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See Theorem 1.1 p. 3.
Sam Buss and Ryan Williams, Limits on alternation-trading proofs for time-space lower bounds, Electronic Colloquium on Computational Complexity 2011
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], Oct 03 2012.
Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Ryan Williams, Time-space tradeoffs for counting NP solutions modulo integers, Computational Complexity 17 (2008), pp. 179-219.
Index entries for algebraic numbers, degree 3

Cf. A039921 (continued fraction).
Cf. A047336, A047341, A116425, A255240, A255249.
Cf. A003558 (the constant is cyclic with period 3, for N = 7).

R:= RealField(200); Reverse(Intseq(Floor(10^110*2*Cos(Pi(R)/7)))); // Marius A. Burtea, Nov 13 2019

evalf(2*cos(Pi/7), 100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[2 Cos[Pi/7], 10, 111][[1]] (* Robert G. Wilson v, Jun 11 2013 *)

default(realprecision, 20080); x=2*cos(Pi/7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b160389.txt", n, " ", d));

Equals 2*A073052. - Michel Marcus, Nov 21 2013
Equals (Re((-(4*7)*(1 + 3*sqrt(3)*i))^(1/3)) + 1)/3, with the real part Re, and i = sqrt(-1). - Wolfdieter Lang, Feb 24 2015
Equals i^(2/7) - i^(12/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 20 2021: (Start)
Equals 2 - (1 - z)*(1 - z^6)/((1 - z^3)*(1 - z^4)), where z = exp(2*Pi*i/7).
The other two zeros of the minimal polynomial x^3 - x^2 - 2*x + 1 of 2*cos(Pi/7) are given by 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)) = 2*cos(3*Pi/7) = A255241 and 2 - (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) = cos(5*Pi/7) = -A362922.
Equals Product_{n >= 0} (7*n+2)*(7*n+5)/((7*n+1)*(7*n+6)) = 1 + Product_{n >= 0} (7*n+2)*(7*n+5)/((7*n+3)*(7*n+4)) = 1/A255240.
The linear fractional mapping r -> 1/(1 - r) cyclically permutes the three zeros of the minimal polynomial x^3 - x^2 - 2*x + 1. The inverse mapping is r -> (r - 1)/r.
The quadratic mapping r -> 2 - r^2 also cyclically permutes the three zeros. The inverse mapping is r -> r^2 - r - 1. (End)
Equals i^(2/7) + i^(-2/7). - Gary W. Adamson, Feb 11 2022
From Amiram Eldar, Nov 22 2024: (Start)
Equals Product_{k>=1} (1 - (-1)^k/A047336(k)).
Equals 1 + cosec(3*Pi/14)/2 = 1 + Product_{k>=1} (1 + (-1)^k/A047341(k)). (End)
Equals sqrt(A116425). - Hugo Pfoertner, Nov 22 2024

A255249 Decimal expansion of -2cos(5Pi/7).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Mar 13 2015

rho_3 := +2*cos(5*Pi/7) is the negative zero of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7) = 2*cos(Pi/7), the length ratio of the smaller diagonal and the side in the regular 7-gon (heptagon). See A187360 and a link to the arXiv paper given there, eq. (20) for the zeros of C(n, x). The positive zeros are rho(7) and rho_2 = 2*cos(3*Pi/7) shown in A160389 and A255241.

Essentially the same as A231187 and A116425. - R. J. Mathar, Mar 14 2015

			1.2469796037174670610500097680084796212645494617928042107310988781937073049...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Wikipedia, Gaussian period.
Index entries for algebraic numbers, degree 3

Cf. A047385, A160389, A187360, A255241, A330002, A330003 (Beatty sequences), A362922.

r = x /. FindRoot[1/x + 1/(x+1)^2 == 1, {x, 2, 10}, WorkingPrecision -> 210]
RealDigits[r][[1]]
Plot[1/x + 1/(x+1)^2, {x, 1, 2}] (* Clark Kimberling, Jan 04 2020 *)

polrootsreal(x^3 + x^2 - 2*x - 1)[3] \\ Charles R Greathouse IV, Oct 30 2023

2*cos(5*Pi/7) = - 2*sin(3*Pi/14) = -1.246979603...
Solution of x^3 + x^2 - 2 x - 1 = 0; +1.246979603...  - Clark Kimberling, Jan 04 2020
Equals i^(4/7) - i^(10/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 20 2021: (Start)
Equals z + z^6, where z = exp(2*Pi*i/7), so this constant is one of the three cubic Gaussian periods for the modulus 7. The other periods are - A255241 and - A160389.
Equals (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) - 2.
Equals Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+2)*(7*n+5)) = A231187 - 1. (End)
Equals Product_{k>=1} (1 - (-1)^k/A047385(k)). - Amiram Eldar, Nov 22 2024
Equals 1/(A160389-1) = 2*A362922. - Hugo Pfoertner, Nov 22 2024

A255241 Decimal expansion of 2cos(3Pi/7).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 13 2015

This is also the decimal expansion of 2*sin(Pi/14).
rho_2 := 2*cos(3*Pi/7) and rho(7) := 2*cos(Pi/7) (see A160389) are the two positive zeros of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7), the length ratio of the smaller diagonal and the side in the regular 7-gon (heptagon). See A187360 and a link to the arXiv paper given there, eq. (20) for the zeros of C(n, x). The other zero is negative, rho_3 := 2*cos(5*Pi/n). See -A255249.
Also the edge length of a regular 14-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272487). - Stanislav Sykora, May 01 2016

			0.445041867912628808577805128993589518932711137529089910623974031...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 3

Cf. A004169, A160389, A187360, A255249, A232736, A255240.

Edge lengths of other nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A130880 (n=18), A272491 (n=19). - Stanislav Sykora, May 01 2016

R:= RealField(120); 2*Cos(3*Pi(R)/7); // G. C. Greubel, Sep 04 2022

RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/14)
```

polrootsreal(x^3 - x^2 - 2*x + 1)[2] \\ Charles R Greathouse IV, Oct 30 2023

numerical_approx(2*cos(3*pi/7), digits=120) # G. C. Greubel, Sep 04 2022

2*cos(3*Pi/7) = 2*sin(Pi/14) = 2*A232736 = 1/A231187 = 0.4450418679...
See A232736 for the decimal expansion of cos(3*Pi/7) = sin(Pi/14).
Equals i^(6/7) - i^(8/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 11 2021: (Start)
Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7).
Equals 1 - A255240. (End)

Offset corrected by Stanislav Sykora, May 01 2016

A255240 Decimal expansion of 1/(2*cos(Pi/7)).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 12 2015

This is the decimal expansion of t = 1/rho(7) = 2 + rho(7) - rho(7)^2 with rho(7) = 2*cos(Pi/7) the length ratio of the smaller diagonal and the side of a regular heptagon. See A160389 for the decimal expansion of rho(7).
t satisfies the cubic equation t^3 - 2*t^2 - t + 1 = 0.
t = 1/rho(7) is the slope tan(alpha) appearing in Archimedes's neusis construction of the regular heptagon. The corresponding angle alpha is approximately 29,028 degrees. See the link, Figure 1, also for references.
From Peter Bala, Oct 16 2021: (Start)
t = sin(Pi/7)/sin(2*Pi/7). The other roots of the cubic equation t^3 - 2*t^2 - t + 1 = 0 are t_1 = 1/(1 - t) = sin(3*Pi/7)/sin(6*Pi/7) = 2.2469796037... and t_2 = 1/(1 - t_1) = - sin(2*Pi/7)/sin(4*Pi/7) = - 0.8019377358.... Compare with A231187 and A160389.
The algebraic number field Q(t) is a totally real cubic field of discriminant 7^2 and class number 1 with a cyclic Galois group over Q of order 3.  See Shanks. (End)

			0.5549581320873711914221948710064104810672888624709100893760259682051575359...

Wolfdieter Lang, Archimedes's Construction of the Regular Heptagon.
Daniel Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
Index entries for algebraic numbers, degree 3.

Cf. A047385, A116425, A160389, A231187, A255241.

RealDigits[1/(2*Cos[Pi/7]), 10, 100][[1]] (* Georg Fischer, Apr 04 2020 *)

1/rho(7) = 1/(2*cos(Pi/7)) = 0.55495813208...
From Peter Bala, Oct 10 2021: (Start)
t = 2*(cos(Pi/7) - cos(2*Pi/7)); t_1 = 2*(cos(3*Pi/7) - cos(6*Pi/7)); t_2 = 2*(cos(5*Pi/7) - cos(10*Pi/7)).
t = Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+2)*(7*n+5)) = 1 - Product_{n >= 0} (7*n+1)*(7*n+6)/((7*n+3)*(7*n+4)) =  1 - A255241. (End)
Equals Product_{k>=1} (1 + (-1)^k/A047385(k)). - Amiram Eldar, Nov 22 2024

Name corrected by Georg Fischer, Apr 04 2020

A116425 Decimal expansion of 2 + 2cos(2Pi/7).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 15 2006

A root of the equation x^3 - 5*x^2 + 6*x - 1 = 0. - Arkadiusz Wesolowski, Jan 13 2016

The other two roots of this minimal polynomial of the present algebraic number (rho(7))^2, with rho(7) = 2*cos(Pi/7) = A160389 are (2*cos(3*Pi/7))^2 = (A255241)^2 and (2*cos(5*Pi/7))^2 = (-A255249)^2. - Wolfdieter Lang, Mar 30 2020

			3.246979603717467061...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.25 Tutte-Beraha Constants, p. 417.

Jesús Salas and Alan D. Sokal, Transfer matrices and partition functions zeros for antiferromagnetic Potts models, arXiv:cond-mat/0004330 [cond-mat.stat-mech], 2000-2001, p. 64.
Eric Weisstein's World of Mathematics, Logistic Map
Eric Weisstein's World of Mathematics, Silver Constant
Index entries for algebraic numbers, degree 3

Cf. A231187, A160389.
Cf. A003558, A054142, A255241, A255249.
2 + 2*cos(2*Pi/n): A104457 (n = 5), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

First@ RealDigits[N[2 + 2 Cos[2 Pi/7], 120]] (* Michael De Vlieger, Jan 13 2016 *)

2 + 2*cos(2*Pi/7) \\ Michel Marcus, Jan 13 2016

Equals (2*cos(Pi/7))^2 = (A160389)^2.
Equals 2 + i^(4/7) - i^(10/7). - Peter Luschny, Apr 04 2020
Let  c = 2 + 2*cos(2*Pi/7). The linear fractional transformation z -> c - c/z has order 7, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/z)))))). - Peter Bala, May 09 2024

A358181 Decimal expansion of the real root of x^3 - 2*x^2 - x - 1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 07 2022

This equals r0 + 2/3 where r0 is the real root of y^3 - (7/3)*y - 61/27.
The other roots of x^3 - 2*x^2 - x - 1 are (2 + w1*((61 + 9*sqrt(29))/2)^(1/3) + w2*((61 - 9*sqrt(29))/2)^(1/3))/3 = -0.2734091384... + 0.5638210928...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (2 - sqrt(7)*(cosh((1/3)*arccosh((61/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((61/98)*sqrt(7)))*i))/3, and its complex conjugate.

			2.5468182768840820791359975088097915288112703374520061295514745747111979831...

Cf. A088559, A231187, 1 - A160389, A255240.

RealDigits[x /. FindRoot[x^3 - 2*x^2 - x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
RealDigits[Root[x^3-2x^2-x-1,1],10,120][[1]] (* Harvey P. Dale, Mar 30 2025 *)

r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + 7*((61 + 9*sqrt(29))/2)^(-1/3))/3.
r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + ((61 - 9*sqrt(29))/2)^(1/3))/3.
r = 2*(1 + sqrt(7)*cosh((1/3)*arccosh((61/98)*sqrt(7))))/3.
r = (2/3) +(2^(2/3)*61^(1/3))/3*Hyper2F1([-1/6,1/3],[1/2],2349/3721). - Gerry Martens, Nov 08 2022

A262770 A Beatty sequence: a(n)=floor(np) where p=2cos(Pi/7)=A160389.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 46, 48, 50, 52, 54, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 109, 111, 113, 115, 117, 118, 120, 122, 124, 126, 127, 129, 131, 133, 135, 136, 138, 140, 142, 144, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 163, 165, 167, 169, 171, 172, 174, 176, 178, 180, 181, 183, 185, 187, 189, 191

Offset: 0

Patrick D McLean, Sep 30 2015

nonn

Beatty sequence of the shorter diagonal (A160389) in a regular heptagon with sidelength 1.
Complement of Beatty sequence A262773 of the longer diagonal (A231187) in a regular heptagon with sidelength 1.
First 106 terms agree with A187318, but A187318(106)=190 while A262770(106)=191.

Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Index entries for sequences related to Beatty sequences

Complement of A262773.

Initially agrees with A187318 (because 2*cos(Pi/7) is close to 9/5).

Table[Floor[2 n Cos[Pi/7]], {n, 0, 106}] (* Michael De Vlieger, Oct 05 2015 *)

p=roots([1,-1,-2,1])(1); a(n)=floor(p*n)

a(n) = floor(n*2*cos(Pi/7)); \\ Michel Marcus, Oct 05 2015

A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2

Offset: 4

Mohammed Yaseen, Aug 22 2020

			Triangle begins:
1;
2, 2;
2, 2, 2;
2, 2, 2, 2;
2, 2, 3, 2, 2;
2, 3, 3, 3, 3, 2;
2, 3, 3, 3, 3, 3, 2;
2, 3, 3, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 4, 3, 3, 2;
2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2;
...
Row n lists the closest integers to the length of the diagonals drawn from a fixed vertex of a regular n-gon with unit edge length, n >= 4.
The lengths of the diagonals drawn from vertex A of a regular 8-gon ABCDEFGH with unit edge length are:
AC = 1.84775...
AD = 2.41421...
AE = 2.61312...
AF = 2.41421...
AG = 1.84775...
So the row for n=8 is 2, 2, 3, 2, 2.

Cf. A064313.
Decimal expansion of diagonal lengths of regular n-gons with unit edge length:
n=4  A002193.
n=5  A001622.
n=6  A002194, A000038.
n=7  A160389, A231187.
n=8  A179260, A014176, A121601.
n=9  A332437.
n=10 A188593, A104457, A019970, A134945.
n=11 A231186.
n=12 A188887, A090388, A337402, A019973, A214726.

T[n_,k_]:=Round[Sin[(k+1)*Pi/n]/Sin[Pi/n]]; Flatten[Table[T[n,k],{n,4,16},{k,1,n-3}]] (* Stefano Spezia, Sep 07 2020 *)

T(n,k) = round(sin((k+1)*Pi/n)/sin(Pi/n)), n >= 4, 1 <= k <= n-3.

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A262773 A Beatty sequence: a(n)=floor(q*n) where q=A231187.

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A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).

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A160389 Decimal expansion of 2*cos(Pi/7).

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A255249 Decimal expansion of -2*cos(5*Pi/7).

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A255241 Decimal expansion of 2*cos(3*Pi/7).

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A255240 Decimal expansion of 1/(2*cos(Pi/7)).

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A255249 Decimal expansion of -2cos(5Pi/7).

A255241 Decimal expansion of 2cos(3Pi/7).

A116425 Decimal expansion of 2 + 2cos(2Pi/7).

A262770 A Beatty sequence: a(n)=floor(np) where p=2cos(Pi/7)=A160389.