OFFSET
1,1
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 10 ).
LINKS
Bruno Berselli, Table of n, a(n) for n = 1..10000 (From Bruno Berselli, Jun 24 2010)
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 (1,1,-1).
FORMULA
a(n) = sqrt(2)*sqrt((1-6*n)*(-1)^n + 18*n^2 - 6*n + 1)/2. - Paul Barry, May 11 2003
From Bruno Berselli, Jun 24 2010: (Start)
G.f.: (3+2*x+x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0, with n > 3.
a(n) = (6*n - (-1)^n - 1)/2. (End)
a(n) = 6*n - a(n-1) - 4 with n > 1, a(1)=3. - Vincenzo Librandi, Aug 05 2010
From Guenther Schrack, Nov 17 2018: (Start)
a(n) = a(n-2) + 6 for n > 2.
a(-n) = -A047241(n+1) for n > 0.
a(n) = A109613(n-1) + 2*n for n > 0.
a(n) = 2*A001651(n) + 1.
m-element moving averages: Sum_{k=1..m} a(n-m+k)/m = A016777(n-m/2) for m = 2, 4, 6, ... and n >= m. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) - log(3)/4. - Amiram Eldar, Dec 13 2021
E.g.f.: 1 + 3*x*exp(x) - cosh(x). - David Lovler, Aug 25 2022
MATHEMATICA
Select[Range@ 149, MemberQ[{3, 5}, Mod[#, 6]] &] (* or *)
Array[(6 # - (-1)^# - 1)/2 &, 50] (* or *)
Fold[Append[#1, 6 #2 - Last@ #1 - 4] &, {3}, Range[2, 50]] (* or *)
CoefficientList[Series[(3 + 2 x + x^2)/((1 + x) (1 - x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jan 12 2018 *)
PROG
(PARI) a(n) = (6*n - 1 - (-1)^n)/2 \\ David Lovler, Aug 25 2022
CROSSREFS
Cf. A047235 [(6*n-(-1)^n-3)/2], A047241 [(6*n-(-1)^n-5)/2], A047238 [(6*n-(-1)^n-7)/2]. [Bruno Berselli, Jun 24 2010]
Subsequence of A186422.
From Guenther Schrack, Nov 18 2018: (Start)
Complement: A047237.
First differences: A105397(n) for n > 0.
Partial sums: A227017(n+1) for n > 0.
Elements of odd index: A016945.
Elements of even index: A016969(n-1) for n > 0. (End)
KEYWORD
nonn,easy
AUTHOR
STATUS
approved