%I #78 Jun 13 2024 14:55:40

%S 3,11,19,27,35,43,51,59,67,75,83,91,99,107,115,123,131,139,147,155,

%T 163,171,179,187,195,203,211,219,227,235,243,251,259,267,275,283,291,

%U 299,307,315,323,331,339,347,355,363,371,379,387,395,403,411,419,427

%N a(n) = 8n + 3.

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

%C Also numbers of the form x^2 + y^2 + z^2, where x,y,z are odd integers. - _Alexander Adamchuk_, Dec 01 2006

%C 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

%H Vincenzo Librandi, <a href="/A017101/b017101.txt">Table of n, a(n) for n = 0..5000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H MathOverflow, <a href="https://mathoverflow.net/q/323164">Absolute oscillator in Langton's ant</a>

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = A001969(2*n+1) + A001969(2*n) = A000069(2*n+1) + A000069(2*n). - _Philippe Deléham_, Feb 04 2004

%F G.f.: (3+5*x)/(1-x)^2. - _R. J. Mathar_, Mar 30 2011

%F a(n) = 2*a(n-1) - a(n-2) for n>1. - _Vincenzo Librandi_, May 28 2011

%F a(A002265(n)) = A004442(n) + A004443(n). - _Wesley Ivan Hurt_, Apr 06 2016

%F E.g.f.: exp(x)*(3 + 8*x). - _Stefano Spezia_, Nov 20 2019

%F a(n) = A004767(2*n), for n >= 0. See also A004767(2*n+1) = A004771(n). - _Wolfdieter Lang_, Feb 03 2022

%p A017101:=n->8*n+3: seq(A017101(n), n=0..100); # _Wesley Ivan Hurt_, Apr 06 2016

%t Range[3, 1000, 8] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2011 *)

%t 8*Range[0,80]+3 (* or *) LinearRecurrence[{2,-1},{3,11},80] (* _Harvey P. Dale_, May 04 2023 *)

%o (Magma) [8*n+3: n in [0..60]]; // _Vincenzo Librandi_, May 28 2011

%o (PARI) a(n)=8*n+3 \\ _Charles R Greathouse IV_, Jun 02 2013

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

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_