A047537 Numbers that are congruent to {1, 4, 7} mod 8.

1, 4, 7, 9, 12, 15, 17, 20, 23, 25, 28, 31, 33, 36, 39, 41, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 71, 73, 76, 79, 81, 84, 87, 89, 92, 95, 97, 100, 103, 105, 108, 111, 113, 116, 119, 121, 124, 127, 129, 132, 135, 137, 140, 143, 145, 148, 151, 153, 156, 159

Offset

1,2

Comments

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 27 ).

Links

Table of n, a(n) for n=1..60.

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).

William A. Stein, The modular forms database.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

Equals A097806 * A042965, where A097806 = the pairwise operator and A042965 = numbers not congruent to 2 mod 4. - Gary W. Adamson, Sep 12 2007

From Wesley Ivan Hurt, Jun 09 2016: (Start)

G.f.: x*(1+3*x+3*x^2+x^3)/((x-1)^2*(1+x+x^2)).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = (24*n-12+3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 8k-1, a(3k-1) = 8k-4, a(3k-2) = 8k-7. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (tan(Pi/16)+cot(Pi/16)-1)*Pi/16 = (2*sqrt(2*(2+sqrt(2)))-1)*Pi/16. - Amiram Eldar, Dec 19 2021

Maple

A047537:=n->(24*n-12+3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9: seq(A047537(n), n=1..100); # Wesley Ivan Hurt, Jun 09 2016

Mathematica

Select[Range[200], MemberQ[{1, 4, 7}, Mod[#, 8]]&] (* or *) LinearRecurrence[{1, 0, 1, -1}, {1, 4, 7, 9}, 100] (* Harvey P. Dale, Apr 01 2016 *)

Prog

(Magma) [n : n in [0..150] | n mod 8 in [1, 4, 7]]; // Wesley Ivan Hurt, Jun 09 2016

Crossrefs

Cf. A042965, A097806.

Sequence in context: A310952 A007066 A260395 * A247985 A190438 A189526

Adjacent sequences: A047534 A047535 A047536 * A047538 A047539 A047540

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved