OFFSET
0,1
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..5000
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).
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
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
MAPLE
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 *)
PROG
(Magma) [8*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011
(PARI) a(n)=8*n+3 \\ Charles R Greathouse IV, Jun 02 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved