A047270 Numbers that are congruent to {3, 5} mod 6.

3, 5, 9, 11, 15, 17, 21, 23, 27, 29, 33, 35, 39, 41, 45, 47, 51, 53, 57, 59, 63, 65, 69, 71, 75, 77, 81, 83, 87, 89, 93, 95, 99, 101, 105, 107, 111, 113, 117, 119, 123, 125, 129, 131, 135, 137, 141, 143, 147, 149

Offset: 1

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( 10 ).

This sequence is an interleaving of A016945 with A016969. - Guenther Schrack, Nov 16 2018

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).

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)

Programs

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 *)

PARI

a(n) = (6*n - 1 - (-1)^n)/2 \\ David Lovler, Aug 25 2022

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