A020735 Odd numbers >= 5.

5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset

1,1

Comments

Values of n such that a regular polygon with n sides can be formed by tying knots in a strip of paper. - Robert A. J. Matthews (rajm(AT)compuserve.com)

These polygons fill in many of the gaps left by the Greeks, who were restricted to compass and ruler. Specifically, they make possible construction of the regular 7-sided heptagon, 9-sided nonagon, 11-gon and 13-gon. The 14-gon becomes the first to be impossible by either ruler, compass or knotting.

Continued fraction expansion of 2/(exp(2)-7). - Thomas Baruchel, Nov 04 2003

Pisot sequence T(5,7). - David W. Wilson

Sun conjectures that any member of this sequence is of the form m^2 + m + p, where p is prime. Blanco-Chacon, McGuire, & Robinson prove that the primes of this form have density 1. - Charles R Greathouse IV, Jun 20 2019

References

F. V. Morley, Proc. Lond. Math. Soc., Jun 1923

F. V. Morley, "Inversive Geometry" (George Bell, 1933; reprinted Chelsea Publishing Co. 1954)

Links

Table of n, a(n) for n=1..64.

Ivan Blanco-Chacon, Gary McGuire, and Oisin Robinson, Primes of the form n^2+n+p have density 1 (2017)

Tanya Khovanova, Recursive Sequences

Z. W. Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory 1:1 (2009), pp. 65-76.

Index entries for sequences related to knots

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 2*n + 3.

G.f.: x*(5-3*x)/(1-2*x+x^2). a(n) = 2*a(n-1)-a(n-2). - Colin Barker, Jan 31 2012

Mathematica

Range[5, 131, 2] (* Harvey P. Dale, Aug 11 2012 *)

Prog

(PARI) a(n)=2*n+3 \\ Charles R Greathouse IV, Jul 10 2016

Crossrefs

Subsequence of A005408. See A008776 for definitions of Pisot sequences.

Sequence in context: A049013 A267872 A062545 * A329391 A108144 A241835

Adjacent sequences: A020732 A020733 A020734 * A020736 A020737 A020738

Keyword

nonn,easy,nice

Author

David W. Wilson

Extensions

Entry revised by N. J. A. Sloane, Jan 26 2007

Status

approved