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

Author

David W. Wilson

Keywords

• nonn
• easy
• nice

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, Proceedings of the London Mathematical Society, Jun 1923.
• F. V. Morley, "Inversive Geometry" (George Bell, 1933; reprinted Chelsea Publishing Co. 1954).

Links

• Ivan Blanco-Chacon, Gary McGuire, and Oisin Robinson, Primes of the form n^2+n+p have density 1, arXiv:1707.06014 [math.NT], 2017.
• Tanya Khovanova, Recursive Sequences.
• Zhi-Wei Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009; 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).

Crossrefs

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

Cf. A016825, A028347.

Programs

Mathematica

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

PARI

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

Formula

a(n) = 2*n + 3.

From Colin Barker, Jan 31 2012: (Start)

G.f.: x*(5-3*x)/(1-2*x+x^2).

a(n) = 2*a(n-1) - a(n-2). (End)

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: exp(x)*(2*x + 3) - 3.

a(n) = A016825(n+1)/2 = A028347(n+2) - A028347(n+1). (End)

Extensions

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