A017101 a(n) = 8n + 3.

3, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 99, 107, 115, 123, 131, 139, 147, 155, 163, 171, 179, 187, 195, 203, 211, 219, 227, 235, 243, 251, 259, 267, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 363, 371, 379, 387, 395, 403, 411, 419, 427

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 47 ).

Also numbers of the form x^2 + y^2 + z^2, where x,y,z are odd integers. - Alexander Adamchuk, Dec 01 2006

Conjecture: 2*a(n) is the half-period of oscillation of a Langton's ant colony that is n basic blocks in length. To construct such blocks use a pair of ants facing north at (x,y) and (x+1,y+2) (using Golly's coordinate system). Each successive block is placed 1 cell away from the previous one, i.e., the x coordinate shifts by 3, so we have (x+3k,y) and (x+3k+1,y+2). Also, because of the symmetry of the oscillation pattern, 4*a(n) is the length of the whole period (see MathOverflow link for details). - Mikhail Kurkov, Nov 20 2019

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 247.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 5, 14.
• Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
• Tanya Khovanova, Recursive Sequences
• MathOverflow, Absolute oscillator in Langton's ant
• William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
• William A. Stein, The modular forms database
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000069, A001969, A002265, A004442, A004443, A004767, A004771, A004769, A017077, A017089.

Programs

Magma

[8*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

Maple

A017101:=n->8*n+3: seq(A017101(n), n=0..100); # Wesley Ivan Hurt, Apr 06 2016

Mathematica

Range[3, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
8*Range[0,80]+3 (* or *) LinearRecurrence[{2,-1},{3,11},80] (* Harvey P. Dale, May 04 2023 *)

PARI

a(n)=8*n+3 \\ Charles R Greathouse IV, Jun 02 2013

Formula

a(n) = A001969(2*n+1) + A001969(2*n) = A000069(2*n+1) + A000069(2*n). - Philippe Deléham, Feb 04 2004

G.f.: (3+5*x)/(1-x)^2. - R. J. Mathar, Mar 30 2011

a(n) = 2*a(n-1) - a(n-2) for n>1. - Vincenzo Librandi, May 28 2011

a(A002265(n)) = A004442(n) + A004443(n). - Wesley Ivan Hurt, Apr 06 2016

E.g.f.: exp(x)*(3 + 8*x). - Stefano Spezia, Nov 20 2019

a(n) = A004767(2*n), for n >= 0. See also A004767(2*n+1) = A004771(n). - Wolfdieter Lang, Feb 03 2022