A047385 -id:A047385 - cp's OEIS frontend

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

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

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

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

Original entry on oeis.org

1, 5, 10, 17, 26, 36, 49, 62, 79, 95, 116, 135, 160, 182, 211, 236, 269, 297, 334, 365, 406, 440, 485, 522, 571, 611, 664, 707, 764, 810, 871, 920, 985, 1037, 1106, 1161, 1234, 1292, 1369, 1430, 1511, 1575, 1660, 1727, 1816, 1886, 1979, 2052, 2149, 2225, 2326

Offset: 0

Bruno Berselli, Oct 20 2011 - based on remarks and sequences by Omar E. Pol

For an origin of this sequence, see the numerical spiral illustrated in the Links section.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195020 (vertices of the numerical spiral in link).

Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].

[(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];

Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* Harvey P. Dale, Jun 19 2020 *)

for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));

O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - G. C. Greubel, Aug 19 2017
a(n) = A195020(n) + n + 1.
a(n)   - a(-n-1)  = A047336(n+1).
a(n+1) - a(-n)    = A113804(n+1).
a(n+2) - a(n)     = A047385(n+3).
a(n+4) - a(n)     = A113803(n+4).
a(2*n) + a(2*n-1) = A069127(n+1).
a(2*n) - a(2*n-1) = A016813(n).
a(2*n+1) - a(2*n) = A016777(n+1).
a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).

A113804 Numbers that are congruent to 4 or 10 mod 14.

Original entry on oeis.org

4, 10, 18, 24, 32, 38, 46, 52, 60, 66, 74, 80, 88, 94, 102, 108, 116, 122, 130, 136, 144, 150, 158, 164, 172, 178, 186, 192, 200, 206, 214, 220, 228, 234, 242, 248, 256, 262, 270, 276, 284, 290, 298, 304, 312, 318, 326, 332, 340, 346, 354, 360

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Fourth row of the 7-rowed array A113807. - Giovanni Teofilatto, Oct 26 2009 [crossref added by Wolfdieter Lang, Dec 15 2011]

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047385, A113801, A113802, A113803, A113805, A113806, A008589, A113807.

Select[Range[2,400,2],MemberQ[{4,10},Mod[#,14]]&] (* or *) LinearRecurrence[{1,1,-1},{4,10,18},60] (* Harvey P. Dale, Jan 08 2023 *)

a(n)=7*n-((-1)^n+7)/2 \\ Charles R Greathouse IV, Dec 27 2011

From R. J. Mathar, Aug 13 2008: (Start)
a(n) = 7n - ((-1)^n + 7)/2.
G.f.: 2x*(2 + 3x + 2x^2)/((1-x)^2*(1+x)). (End)
a(n) = 14*n - a(n-1) - 14 (with a(1)=4). - Vincenzo Librandi, Aug 01 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
E.g.f.: 4 + ((14*x - 7)*exp(x) - exp(-x))/2. - David Lovler, Sep 04 2022
a(n) = 2*A047385(n). - Michel Marcus, Sep 05 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(Pi/7)/2.
Product_{n>=1} (1 + (-1)^n/a(n)) = tan(3*Pi/14). (End)

More terms from Neven Juric, Apr 10 2008

A047380 Numbers that are congruent to {1, 2, 4, 5} mod 7.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 12, 15, 16, 18, 19, 22, 23, 25, 26, 29, 30, 32, 33, 36, 37, 39, 40, 43, 44, 46, 47, 50, 51, 53, 54, 57, 58, 60, 61, 64, 65, 67, 68, 71, 72, 74, 75, 78, 79, 81, 82, 85, 86, 88, 89, 92, 93, 95, 96, 99, 100, 102, 103, 106, 107, 109, 110

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002265, A047346, A047385.

A047380:=n->(14*n-11-3*I^(2*n)-(1+I)*I^(-n-1)-(1-I)*I^(n+1))/8: seq(A047380(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Table[(14n-11-3I^(2n)-(1+I)I^(-n-1)-(1-I)I^(n+1))/8, {n, 80}] (* Wesley Ivan Hurt, May 20 2016 *)
LinearRecurrence[{1,0,0,1,-1},{1,2,4,5,8},100] (* Harvey P. Dale, Jun 05 2016 *)

a(n)=(n-1)\4*7+[5,1,2,4][n%4+1] \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(1+x+2*x^2+x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n - 11 - 3*i^(2*n) - (1+i)*i^(-n-1) - (1-i)*i^(n+1))/8 where i=sqrt(-1).
a(2n) = A047385(n), a(2n-1) = A047346(n). (End)

A047315 Numbers that are congruent to {2, 4, 5, 6} mod 7.

Original entry on oeis.org

2, 4, 5, 6, 9, 11, 12, 13, 16, 18, 19, 20, 23, 25, 26, 27, 30, 32, 33, 34, 37, 39, 40, 41, 44, 46, 47, 48, 51, 53, 54, 55, 58, 60, 61, 62, 65, 67, 68, 69, 72, 74, 75, 76, 79, 81, 82, 83, 86, 88, 89, 90, 93, 95, 96, 97, 100, 102, 103, 104, 107, 109, 110, 111

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047288, A047385.

[n : n in [0..150] | n mod 7 in [2, 4, 5, 6]]; // Wesley Ivan Hurt, Jun 03 2016

A047315:=n->(14*n-1-I^(2*n)-(3-I)*I^(-n)-(3+I)*I^n)/8: seq(A047315(n), n=1..100); # Wesley Ivan Hurt, Jun 03 2016

Table[(14n-1-I^(2n)-(3-I)*I^(-n)-(3+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 03 2016 *)
Select[Range[200],MemberQ[{2,4,5,6},Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,0,1,-1},{2,4,5,6,9},100] (* Harvey P. Dale, Jan 19 2019 *)

G.f.: x*(2+2*x+x^2+x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 03 2011
From Wesley Ivan Hurt, Jun 03 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-1-i^(2*n)-(3-i)*i^(-n)-(3+i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047288(k), a(2k-1) = A047385(k). (End)

A047331 Numbers that are congruent to {2, 3, 5, 6} mod 7.

Original entry on oeis.org

2, 3, 5, 6, 9, 10, 12, 13, 16, 17, 19, 20, 23, 24, 26, 27, 30, 31, 33, 34, 37, 38, 40, 41, 44, 45, 47, 48, 51, 52, 54, 55, 58, 59, 61, 62, 65, 66, 68, 69, 72, 73, 75, 76, 79, 80, 82, 83, 86, 87, 89, 90, 93, 94, 96, 97, 100, 101, 103, 104, 107, 108, 110, 111

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047280, A047385.

[n : n in [0..150] | n mod 7 in [2, 3, 5, 6]]; // Wesley Ivan Hurt, Jun 03 2016

A047331:=n->(14*n-3-3*I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/8: seq(A047331(n), n=1..100); # Wesley Ivan Hurt, Jun 03 2016

Table[(14n-3-3*I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 03 2016 *)

G.f.: x*(2+x+2*x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Jun 03 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-3-3*i^(2*n)-(1-i)*i^(-n)-(1+i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047385(k). (End)

A047371 Numbers that are congruent to {0, 2, 3, 5} mod 7.

Original entry on oeis.org

0, 2, 3, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 30, 31, 33, 35, 37, 38, 40, 42, 44, 45, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 79, 80, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 107, 108, 110

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047355, A047385.

[n : n in [0..150] | n mod 7 in [0, 2, 3, 5]]; // Wesley Ivan Hurt, Jun 04 2016

seq(floor((7*n-6)/4), n=1..56); # [Gary Detlefs, Mar 06 2010]

Table[I^(-n)*((14n-15)*I^n+I-1-(1+I)*I^(2n)+I^(-n))/8, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,2,3,5,7},70] (* Harvey P. Dale, Oct 24 2018 *)

a(n) = floor((7n-6)/4). [Gary Detlefs, Mar 06 2010]
G.f.: x^2*(2+x+2*x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = i^(-n)*((14*n-15)*i^n+i-1-(1+i)*i^(2*n)+i^(-n))/8 where i=sqrt(-1).
a(2k) = A047385(k), a(2k-1) = A047355(k). (End)
E.g.f.: (8 + sin(x) - cos(x) + (7*x - 8)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, Jun 04 2016

A047372 Numbers that are congruent to {1, 2, 3, 5} mod 7.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 19, 22, 23, 24, 26, 29, 30, 31, 33, 36, 37, 38, 40, 43, 44, 45, 47, 50, 51, 52, 54, 57, 58, 59, 61, 64, 65, 66, 68, 71, 72, 73, 75, 78, 79, 80, 82, 85, 86, 87, 89, 92, 93, 94, 96, 99, 100, 101, 103, 106, 107, 108, 110

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047356, A047385.

[n : n in [0..150] | n mod 7 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, Jun 04 2016

A047372:=n->(14*n+(3*I-1)*(-I)^n-(3*I+1)*I^n-(-1)^n-13)/8: seq(A047372(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Flatten[7Range[0,12]+n/.n->{1,2,3,5}] (* Harvey P. Dale, Dec 13 2010 *)

From Bruno Berselli, Dec 01 2010: (Start)
G.f.: x*(1+x+x^2+2*x^3+2*x^4) / ((1-x)^2*(1+x+x^2+x^3)).
a(n) = (14*n+(3*i-1)*(-i)^n-(3*i+1)*i^n-(-1)^n-13)/8, i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047385(k), a(2k-1) = A047356(k). (End)
E.g.f.: (8 + 3*sin(x) - cos(x) + (7*x - 6)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, Jun 04 2016

A213904 a(n) is the initial member of the least pair of prime quadruples (of the form p, p+2, p+6, p+8) with a difference of 30*n, with no other prime quadruple between the pair.

Original entry on oeis.org

1006301, 0, 11, 1022381, 0, 3512051, 1871, 632081, 0, 1121831, 15731, 0, 1481, 1155611, 1068251, 0, 18911, 284741, 0, 12390011, 191, 821, 0, 3837131, 875261, 0, 854921, 10865291, 18041, 0, 958541, 680291, 0, 299471, 1063961, 663581, 0, 165701

Offset: 1

Ray G. Opao, Jun 24 2012

nonn

a(n) is 0 if no such pair of prime quadruples is conjectured to exist for the indicated difference.

When n is congruent to 2 or 5 mod 7 (A047385) no solution exists because one of the terms is divisible by 7. [Jud McCranie, Jun 17 2013]

			For n=3, a(3)=11, since 11, 13, 17, 19 is a prime quadruple. The next prime quadruple is 101, 103, 107, 109. The difference 101-11=90, which is equal to 30*3.

Jud McCranie, Table of n, a(n) for n = 1..10000

Cf. A007530, A059925, A157967, A047385.

cp's OEIS Frontend

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

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

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A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

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A113804 Numbers that are congruent to 4 or 10 mod 14.

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A047380 Numbers that are congruent to {1, 2, 4, 5} mod 7.

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A047315 Numbers that are congruent to {2, 4, 5, 6} mod 7.

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A047331 Numbers that are congruent to {2, 3, 5, 6} mod 7.

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

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.